Curves and line bundles on Del Pezzo surfaces
Andreas Leopold Knutsen
We extend the results of Pareschi \cite{par} on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree $1$, where we explicitly classify the cases where the gonality and Clifford index are not constant. We also classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension $3$. Moreover, the property of being exceptional holds for all curves in the complete linear system. Furthermore, we relate the Clifford index and gonality of smooth curves in $|L|$ to the higher order embedding properties of $|L+K_S|$. More precisely, we show that for a nef line bundle $L$ on a Del Pezzo surface, $L+K_S$ is birationally $k$-very ample if and only if all the smooth curves in $|L|$ have gonality $\geq k+2$, and we also find numerical criteria for birational $k$-very ampleness.
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\hfuzz5pc


  %\renewcommand{\baselinestretch}{2}
  \begin{document}

  \title{Curves and line bundles on Del Pezzo surfaces}
  \author{Andreas Leopold Knutsen}  
  
  \address{Universit\"at GH-Essen \\ 
    Fachbereich 6, Mathematik \\ 45 117 Essen\\ Germany}


\email{andreas.knutsen@uni-essen.de}
\keywords{line bundles, curves, Del Pezzo surfaces, gonality,
Clifford index, higher order embeddings}
\subjclass{14J26 (14H51)}
  

\begin{abstract}
We extend the results of Pareschi \cite{par} on the
constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$
to the case of Del Pezzo surfaces of degree $1$, where we explicitly classify the cases where the gonality and Clifford index are not constant. We also classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension $3$. Moreover, the property of being exceptional holds for all curves in the complete linear system.

Furthermore, we relate the Clifford index and gonality of smooth curves in $|L|$ to the higher order embedding properties of $|L+K_S|$. More precisely, we show that for a nef line bundle $L$ on a Del Pezzo surface, $L+K_S$
is birationally $k$-very ample if and only if all the smooth curves in
$|L|$ have gonality $\geq k+2$, and we also find numerical criteria
for birational $k$-very ampleness.
  \end{abstract}

\maketitle

\section{Introduction} \label{intro}

The three main purposes of this paper are (1) to study the gonality and Clifford index of linearly equivalent smooth curves in $|L|$ on a Del Pezzo surface of degree $1$, (2) to relate these concepts, for Del Pezzo surfaces of any degree,  to properties of the line bundle $\O_S(L+K_S)$, more precisely to the {\it birational $k$-very ampleness} of $\O_S(L+K_S)$, a notion we introduced in \cite{kn}, and (3) to classify all cases of exceptional curves on Del Pezzo surfaces.

In the past decades, several authors have studied the question whether exceptional linear series on a curve $C$ on a certain surface propagate to the other members of $|C|$. For instance, for a $K3$ surface, Saint-Donat proved in \cite{S-D} that $C$ possesses a $g^1_2$ or a $g^1_3$ if and only if every smooth curve in $|C|$ does, and Reid \cite{reid} extended this result to other $g^1_d$s. Harris and Mumford conjectured that linearly equivalent smooth curves on a $K3$ surface have the same gonality, but a counterexample of Donagi and Morrison \cite{dm} proved this to be false. This led Green \cite{gre} to modify the conjecture to the effect that linearly equivalent smooth curves on a $K3$ surface have the same Clifford index. This was proved by Green and Lazarsfeld in the famous paper \cite{gl}. Moreover, the Donagi-Morrison counterexample is still the only example known where the gonality is not constant, and Ciliberto and Pareschi \cite{cp} proved that this is indeed the only counterexample if $\O_S(C)$ is ample.

The results of Green and Lazarsfeld were extended to Del Pezzo surfaces of degree $\geq 2$ by Pareschi in \cite{par}. In fact, he showed that for $\deg S \geq 2$ the Clifford index of the smooth curves in a linear 
system $|L|$ containing a smooth curve (which is equivalent to $L$ being nef)
such that $g(L) \geq 4$ is constant, and that the
gonality also is when $g(L) \geq 2$, except for the following case \cite[Example(2.1)]{par}:
\begin{quotation} \textbf{Case (I):} $\deg S = 2$ and  $L \sim -2K_S$ (in particular $g(L)=3$). In this 
case there is a codimension $1$ family of smooth hyperelliptic curves in 
$|L|$, whereas the general smooth curve is trigonal. 

More precisely, denote by $\phi: S \khpil \PP^2$ the morphism defined by $-K_S$, which is a double cover ramified along a smooth quartic. Then
\[ H^0(L) = \phi^{*} H^0(\O_{\PP^2} (2)) \+ W, \]
where $W$ is the $1$-dimensional subspace of sections vanishing on the ramification locus. The smooth curves in the first summand are double covers of conics, whence hyperelliptic, whereas the general curve in the linear system maps isomorphically to a smooth plane quartic and is therefore trigonal.
\end{quotation}

By work of Serrano \cite{ser} there is an example when $\deg S=1$ (namely $L \sim -nK_S$, for $n \geq 3$) where neither the gonality nor the Clifford index is constant (see Example \ref{serrano} below). This shows that the situation is more subtle for $\deg S=1$.

We will classify all the cases for $\deg S=1$ where the gonality and the Clifford index are not constant, which will include the example of Serrano. 

More precisely, we will show that the Clifford index is constant when $\deg S =1$, except precisely for the case (III) below, and that the gonality is constant except precisely for the cases (II) and (III) below:

\begin{quotation} \textbf{Case (II):} $\deg S = 1$ and 
$L \sim -2K_S+2\Gamma$, for a $(-1)$-curve $\Gamma$ (in particular $g(L)=3$). In this case there is also a 
codimension $1$ family of smooth hyperelliptic curves in $|L|$, whereas the 
general smooth curve is trigonal. 
%
%In fact, denoting by $\phi: S \khpil \PP^2$ the morphism defined by 
%$D := -K_S + \Gamma$, which is a double cover ramified along a smooth
%quartic, we get as in Case (I),
%\[ H^0(L) = \phi^{*} H^0(\O_{\PP^2} (2)) \+ W, \]
%where $W$ is the $1$-dimensional subspace of sections vanishing on the
%ramification locus, and the smooth curves in the first summand are 
%hyperelliptic, whereas the general curve in the linear system is trigonal;
\end{quotation}
and 
\begin{quotation} \textbf{Case (III):} 
$\deg S = 1$, $L$ is ample,
  $L.\Gamma \geq 2$ for all $(-1)$-curves $\Gamma$ if $L^2 \geq 8$,
  and there is an integer $d \geq 3$ such that
$L^2 \geq 5d-8 \geq 7$, $L.K_S =-d$,  $L.C \geq d$ for all smooth rational curves $C$ with $C^2 =0$ and  $L.C \geq d+1$ for all smooth rational curves $C$ with $C^2 =1$ if $(d,L^2)=(4,8)$.

In this case there is a codimension $1$ family of $|L|$ of smooth $(d-1)$-gonal curves (which are exactly the smooth curves passing through the base point of $K_S$), whereas the general smooth curve is $d$-gonal.

If $g(L) \geq 4$, then the smooth curves in the codimension $1$ family have Clifford index $d-3$, whereas the general smooth curve has  Clifford index $d-2$.
\end{quotation}

%Unfortunately, we are not able to give a better description of Case
%(III) than the above one. However, by working with the Picard group of
%$S$, one should be able - at least in principle - to classify all the
%line bundles $L$ in Case (III).

The three easiest examples of Case (III) are $L \sim -dK_S$ for $d \geq 3$ (precisely the case
described by Serrano \cite{ser}, see Example \ref{serrano} below and also \cite[Example(2.2)]{par}), 
\linebreak $L \sim -(d-1)K_S + \Gamma$, for $d \geq 3$ and a $(-1)$-curve $\Gamma$, and 
$L \sim -(d-2)K_S + R$, for $d \geq 4$ and a smooth rational curve $R$ with $R^2=0$.

We also study exceptional curves on Del Pezzo surfaces of any degree. We show that such curves are exactly the strict transforms of smooth plane curves (whence of Clifford dimension $2$), and of smooth curves in $|-3K_S|$ for $\deg S=3$ (of Clifford dimension $3$).

\begin{thm} \label{gonconst}
  Let $L$ be a nef line bundle on a Del Pezzo surface $S$ with $g(L) \geq 2$. 

  \begin{itemize}
  \item[(a)] All the smooth
  curves in $|L|$ have the same gonality if and only if $L$ is not as in the 
  cases (I), (II) or (III) above. 
  \item[(b)] If $g(L) \geq 4$, then all the smooth
  curves in $|L|$ have the same Clifford index if and only if $L$ is not as in case (III) above.
  \item[(c)]  If $g(L) \geq 4$, then either all or none of the smooth
  curves in $|L|$ are exceptional, and if they are, they all have the same Clifford index and Clifford dimension. Furthermore, the curves are exceptional of Clifford dimension $r \geq 2$ and Clifford index $c$ if and only if $L$ is as in one of the two following cases:
\begin{itemize}
\item[(i)] There exists a base point free line bundle $D$ on $S$
  satisfying  
  \begin{eqnarray*}
    & D^2=1, \hs D.K_S=-3, \hs D.L=c+4 \hs 
              \mbox{ and } \\
\nonumber  &  \Gamma.C \leq 1 \hs \mbox{ for any $(-1)$-curve $\Gamma$ such that $\Gamma.D=0$}
  \end{eqnarray*}
 (the latter condition is equivalent to $D_C$ being very ample for all $C \in |L|$). 

In this case $r=2$, the Clifford dimension is computed by $\O_C(D)$ for all smooth curves $C \in |L|$ and $\phi_D:S \khpil \PP^2$ gives the identification $S \iso S_{m(D)}$ and maps every smooth curve in $|L|$ to a smooth plane
  curve of degree $D.L=c+4$, whence all these curves are exceptional
  of Clifford dimension $2$ and Clifford index $c$.
  \item[(ii)] $S \iso S_n$ for $n \in \{6,7,8 \}$ and 
\[ L \sim -3K_{S_n} + \sum _{i=1}^{n-6} a_i \Gamma_i, \hs \Gamma_i=(-1)\mbox{-curve}, \hs a_i \in \{2,3 \}. \]

In this case $r=3$, $g(L)=10$, $c=3$ and for all smooth curves $C \in |L|$, the Clifford dimension is computed by $\O_C(D)$, for $D:= -K_{S_n} + \sum _{i=1}^{n-6} \Gamma_i$.
\end{itemize}  
\end{itemize}  

\end{thm}

We actually prove a more precise result. In fact, any line bundle $A$ computing the gonality of a smooth curve on a Del Pezzo surface, is of a particular 
form (it is, so to speak, induced from a line bundle on the surface), and we are able to find all possible such in Proposition \ref{allpencils} below.

The second main topic in this paper is the theory of higher order embeddings of surfaces. This theory developped quickly in the past ten years and the $k$-very ampleness (see definition in Section \ref{kvery}) of line bundles on particular kinds of surfaces has been intensively studied in e.g. \cite{ba}, \cite{BFS}, \cite{BS1}, \cite{BS2}, \cite{BS3}, \cite{BS4}, \cite{dr}, \cite{dr2} and \cite{ter}.

In \cite{kn} we introduced the notion of birational $k$-very
ampleness of a line bundle $L$ on an algebraic surface (see Definition \ref{birdef} below) and gave relations
between this notion and the Clifford index and gonality of the
smooth curves in $|L|$ when the surface is $K3$, thus relating the theory of higher order embeddings to the question of propagating linear series on curves studied by the above mentioned authors. We also posed the
question whether there are similar relations for other surfaces
\cite[Rem. 1.5]{kn}.

In this paper we find that there are indeed such relations on a Del
Pezzo surface. 

Let $L$ be a nef line bundle on a Del Pezzo surface (i.e. $L$ is base point 
free or $L \sim -K_{S_8}$ by \cite{dr}). Define the set of curves $\R(L)$  by:
\[ \R (L) := \{ \Gamma \hs | \hs \Gamma=(-1)\mbox{-curve} \hs \mbox{and} \hs\Gamma.L = 0 \}. \]
Furthermore, denote by $m(L)$ the cardinality of $\R (L)$.

Note that by the Hodge index theorem, the curves in $\R(L)$ are necessarily 
disjoint.

The second main result of this paper is the following:

\begin{thm} \label{birkvathm}
  Let $L$ be a nef line bundle of sectional genus $g(L) \geq 2$ on a Del 
  Pezzo surface $S$ and $k \geq 1$ an integer. 
  The following conditions are equivalent:
\begin{itemize}
  \item [(a)] $L+K_S$ is birationally $k$-very ample,
  \item [(b)] $L+K_S$ is birationally $k$-spanned,
  \item [(c)] $L.D \geq k+2$ (resp. $L.D \geq k+3$) for all smooth rational curves $D$ with $D^2=0$ (resp. $D^2=1$) and $-K_S.L \geq k+2+m(L)+K_S^2$ whenever $K_S^2+m(L) \leq \min \{ k+1,4 \} $,
  \item [(d)] all the smooth curves in $|L|$ have gonality $\geq k+2$.
\end{itemize}
If in addition $L +K_S$ is base point free (which happens if and only if $L$ is ample and $L \not \sim -2K_{S_8}$), so that it defines a morphism $\phi_{L+K_S}: S \khpil \PP^{g(L)-1}$, then the following condition is also equivalent to the others:
\begin{itemize}
\item[(e)]  $\phi_{L+K_S} (S)$ has no $(k+1)$-secant $(k-1)$-planes except possibly for those containing the images of $(-1)$-curves $\Gamma$ with $\Gamma.L \leq k$. 
\end{itemize}
Furthermore, if any of these conditions are satisfied, we must have 
$g(L) \geq 2k+1$ (and consequently $L^2 \geq 4k+3$).
\end{thm}

We note that the overall pattern here is that more ``special'' examples turn up the lower the degree of $S$ is (equivalently, the more points of $\PP^2$ we blow up). In fact, up to $\deg S=4$, all linearly equivalent smooth curves have the same gonality and Clifford index and the only exceptional curves are the smooth plane curves. For $\deg S=3$, new examples of exceptional curves turn up, for $\deg S=2$, the first example of linearly equivalent smooth curves of different gonalities appears, and for $\deg S=1$, more such examples turn up, together with examples of linearly equivalent smooth curves of different Clifford indices.

It would be an interesting problem, to study the picture when one blows up $\geq 9$ points of $\PP^2$ (in general position), to see which new examples might turn up. Unfortunately, the methods in this paper rely upon $-K_S$ being ample (or at least effective) and at the moment we do not see how to treat higher blow ups of $\PP^2$.

The paper is organised as follows. 

First, in Section \ref{back} we gather basic definitions that will be needed throughout the paper.

In Section \ref{bir} we treat birational $k$-very ampleness of line bundles and gonality of curves on Del Pezzo surfaces. In particular we relate these concepts to the existence of certain types of line bundles on the surface. We finish the proof of Theorem \ref{birkvathm} at the end of the section, and as examples we state the results for $k=1$ and $k=2$ in Propositions \ref{easyapp} and  
\ref{easyappII}.

In Section \ref{constgon} we show that the gonality is constant for all the smooth curves in a complete linear system on a Del Pezzo surface, except for three special cases. We prove that two of these cases are exactly Cases (I) and (II) given above. We prove that the third case is Case (III) in Section \ref{conc}.

We classify all exceptional curves on Del Pezzo surfaces in Section \ref{exc}, thus finishing the proof of Theorem \ref{gonconst}(c). The methods are similar to the ones used by Pareschi \cite{par}, but due to the extension to Del Pezzo surfaces of degree $1$, we need a more careful analysis. We also use results about exceptional curves in \cite{elms}.

The above mentioned treatment of Case (III) in Section \ref{conc} then concludes the proof of parts (a) and (b) of Theorem \ref{gonconst}.

Finally, in Section \ref{pencilssection} we give the complete list of all the possible pencils computing the gonality of a smooth curve on a Del Pezzo surface. 

Note that parts (a) and (b) of Theorem \ref{gonconst} for $\deg S \geq 2$ are equal to the results of Pareschi \cite{par}. In our proofs, we however never assume that $\deg S =1$, both to give a more complete exposition, and also because restricting to the case $\deg S=1$ would not make the proofs easier, since the case $\deg S=1$ is exactly the case where most problems show up.

\begin{rem}
  {\rm The smooth curves in case (II) above are simply the strict transforms of smooth curves in $|-2K_S|$ for $\deg S=2$ which do not pass through the blown up point.

  The strict transforms of a smooth curve passing through the blown up point are the curves in case (III) with $L \sim 2K_S+\Gamma$.}
\end{rem}


\section{Background material} \label{back}

\subsection{Basic notation and definitions}
The ground field is the field of complex numbers. All surfaces are \textit{smooth irreducible algebraic surfaces}.

By $B_{P_1,...,P_n}(S)$ we will mean the blowing up of $S$ at the points $P_1,...,P_n$

By a \textit{curve} on  a surface $S$ is always meant an \textit{irreducible curve} (possibly singular), i.e. a prime divisor. Line bundles and divisors are used with little or no distinction, as well as the multiplicative and additive notation. Linear equivalence of divisors is denoted by $\sim$.

If $L$ is any line bundle on a surface, $L$ is said to be \textit{numerically effective}, or simply \textit{nef}, if $L.C \geq 0$ for all curves $C$ on $S$. In this case $L$ is said to be \textit{big} if $L^2 >0$.

If $\F$ is any coherent sheaf on a variety $V$, we shall denote by 
$h^i (\F)$ the complex dimension of $H^i (V, \F)$, and by $\chi (\F)$ the Euler characteristic $\sum (-1)^i h^i (\F)$.

The \textit{sectional genus} of a line bundle $L$ is the integer $g(L)=\frac{1}{2}L.(L+K_S)+1$. If $D \in |L|$, then $g(L)$ is the arithmetic genus of $D$.

If $C$ is a curve on a surface and $L$ is a line bundle on the surface, we often use the notation $L_C := L \* \O_C$.

\subsection{$k$-very ampleness} \label{kvery}

A line bundle $\L$ on a smooth variety $X$ is said to be $k$-very ample (resp. $k$-spanned), for an integer $k \geq 0$, if for each (resp. each curvilinear) $0$-dimensional subscheme $\Z$ of $X$ of length $h^0(\O_{\Z})=k+1$, the restriction map $H^0 (L) \khpil H^0 (L \* \O_{\Z})$ is surjective.

We will need the following result:

\begin{thm} \label{bs12}
  \cite[(1.2)]{BS1} If $L$ is a special $k$-spanned line bundle on
  a smooth curve of genus $g$, then $g \geq 2k+1$.
\end{thm}


In \cite{kn} we made the following definition:

\begin{defn}  \label{birdef}
  Let $L$ be a line bundle on a smooth surface 
  $S$ and $k \geq 1$ an integer. 

  $L$ is \textit{birationally $k$-very ample} (resp. \textit{birationally 
  $k$-spanned)}, if there exists a non-empty Zariski-open 
  subset of $S$ where $L$ is $k$-very ample (resp. $k$-spanned).
\end{defn}


\subsection{Del Pezzo surfaces}

A surface $S$ is called a Del Pezzo surface if its anticanonical bundle $-K_S$ is ample. The degree of $S$ is defined as $\deg S = K_S^2$. We have the following classification of such surfaces:

\begin{prop} \label{class}
  \cite{dem} Let $S$ be a Del Pezzo surface. Then
\begin{itemize}
  \item[(a)] $1 \leq \deg S \leq 9$,
  \item[(b)] $\deg S=9$ if and only if $S \iso \PP^2$,
  \item[(c)] if $\deg S=8$, then either $S \iso \PP^1 \x \PP^1$ or 
            $S \iso B_{P}(\PP^2)$,
  \item[(d)] if $1 \leq \deg S \leq 7$, then 
            $S \iso B_{P_1,...,P_{9-\deg S}}(\PP^2)$.
\end{itemize}
\end{prop}

We will often denote a Del Pezzo surface of degree $\deg S$ by $S_n$, where $n=9-\deg S$. Let $\pi: S \khpil \PP^2$ be the blowing up as in (d). Denote by $l$ the class of $\pi ^* (\O_{\PP^2}(1))$ and by $e_i$ the class of 
$\pi ^{-1}(P_i)$. We then have 
\[ l^2=1, \hs e_i.e_j = -\delta_{ij}, \hs e_i.l=0, \]
and 
\[ \Pic S_n \iso \ZZ l \+ \ZZ e_1 \+...\+ \ZZ e_n \iso \ZZ^{n+1}. \]
Therefore any line bundle on $S_n$ is of the form $al - \sum_1^n b_ie_i$ and 
$-K_S = 3l - \sum_1^n e_i$. 

We gather some basic properties of line bundles on a Del Pezzo surface:
\begin{itemize}
\item[(P1)] $|-K_S|$ is base point free unless $S=S_8$, in which case it has exactly one base point, namely the ninth point of intersection of two independent plane cubics based on $9$ points.
\item[(P2)] $|-K_S|$ is very ample if and only if $\deg S \geq 3$. 
\item[(P3)] If $C$ is a connected reduced curve, then $h^1(-C)=h^1(C+K_S)=0$ 
     \cite[(0.4.5)]{par}. 
\item[(P4)] $L$ is nef if and only if it is base point free or 
     $L \sim -K_{S_8}$ \cite[(Cor. 4.7)]{dr}.
\item[(P5)] If $L$ is base point free, then $h^1(L)= h^2(L)=0$ 
     \cite[(0.4.6)]{par}. 
\item[(P6)] If $L >0$ and $L \not \sim -K_{S_8}$, then $L.K_S \geq 2$ (Hodge index theorem and adjunction formula).
\end{itemize}

In particular, if $L$ is nef with $g(L) \geq 1$ we have $h^i(L)=h^i(L+K_S)=0$, for $i=1,2$, $h^0(L+K_S) = g(L) \geq 1$ and the general member of $|L|$ is smooth and irreducible (by Bertini).

By a $(-1)$-curve on a Del Pezzo surface is meant a curve $\Gamma$ such that $\Gamma ^2=-1$. By the adjunction formula, such a curve is necessarily smooth and rational and satisfies $\Gamma.K_S=-1$. The number of such curves on a Del Pezzo surface is finite by standard arguments. Also note that those are the only curves with negative self-intersection on a Del Pezzo surface. It follows that if a line bundle $D$ is not nef, then there exists a $(-1)$-curve $\Gamma$ such that $\Gamma.D <0$.










\subsection{Clifford index, gonality and the vector bundles $E(C,A)$}
\label{cliff}
 
We briefly recall the definition and some properties of gonality and Clifford 
index of curves.
Let $C$ be a smooth irreducible curve of genus $g \geq 2$.
We denote by $g^r_d$ a linear system of dimension $r$ and degree $d$ and say 
that $C$ is $k$-{\it gonal} (and that $k$ is its {\it gonality}) if $C$
posesses a $g^1_k$ but no $g^1_{k-1}$. In particular, we call a $2$-gonal 
curve {\it hyperelliptic} and a $3$-gonal curve {\it trigonal}. We denote by $\gon C$ the gonality of $C$. Note that if 
$C$ is $k$-gonal, all $g^1_k$'s must necessarily be base point free and 
complete.

If $g \geq 4$ and $A$ is a line bundle
on $C$, then the {\it Clifford index} of $A$ is the integer
\[ \Cliff A = \deg A - 2(h^0 (A) -1). \]
The {\it Clifford index of $C$} itself is defined as 
\[ \Cliff C = \min \{ \Cliff A \hspace{.05in} | \hspace{.05in} h^0 (A) \geq 2, h^1 (A) \geq 2 \}. \]
Clifford's theorem then states that $\Cliff C \geq 0$ with equality if
and only if $C$ is hyperelliptic and $\Cliff C =1$ if
and only if $C$ is trigonal or a smooth plane quintic.

At the other extreme, we get from Brill-Noether theory (cf.
\cite[V]{acgh}) that the gonality of $C$ satisfies $\gon C \leq \lfloor 
  \frac{g+3}{2} \rfloor $, whence $\Cliff C \leq \lfloor 
  \frac{g-1}{2} \rfloor $. For the general curve of genus
  $g$, we have $\Cliff C = \lfloor \frac{g-1}{2} \rfloor $.

We say that a line bundle $A$ on $C$ {\it contributes to the Clifford
  index of $C$} if $h^0 (A), h^1 (A) \geq 2$ and that it {\it computes 
the Clifford index of $C$} if in addition $\Cliff C = \Cliff A$. 

Note that $\Cliff A = \Cliff (\omega _C \* A^{-1})$ and also observe
that by Riemann-Roch
\begin{equation}
  \label{eq:007}
h^0(A) + h^1(A)= g+1- \Cliff A.   
\end{equation}

The {\it Clifford dimension} of $C$ is defined as
\[ \min \{ h^0 (A) -1 \mbox{  } | \mbox{  } A  
\mbox{ computes the Clifford index of } C \} . \]

A line bundle $A$ which achieves the minimum and computes the Clifford index, is said to {\it compute} the Clifford dimension. A curve of Clifford index $c$ is $(c+2)$-gonal if and only if it has Clifford dimension $1$. For a general curve $C$, we have  $\gon C = c+2$.

\begin{lemma} \label{gon1}
  \cite[Theorem 2.3]{cm} The gonality $k$ of a smooth irreducible projective curve $C$ satisfies 
\[ \Cliff C +2 \leq k \leq \Cliff C +3.\]
\end{lemma}

The curves satisfying $\gon C = \Cliff C +3$ are conjectured to be very rare and called {\it exceptional} (cf. \cite[(4.1)]{ma}).

We have the following interesting connection between the gonality of a curve and the higher order very ampleness of $\omega_C$:


\begin{lemma} \label{gonkva}
  \cite[Prop. 1.2]{mp} A smooth irreducible projective curve $C$ og genus $g \geq 1$ is $k$-gonal if and only if $\omega_C$ is $(k-2)$-very ample but not $(k-1)$-very ample.
\end{lemma}

Let $C$ be a smooth curve on a regular surface $S$ (i.e. $h^1 (\O_S)=0$). 
Recall that if $A$ is a line 
bundle on $C$ which is generated by its global sections, then one can associate to the pair 
$(C,A)$ a vector bundle $E(C,A)$ of rank $h^0 (A)$ as follows (this vector bundle was introduced by Lazarsfeld in \cite{la}, for more details we 
refer to \cite{cp} and \cite{par}). Thinking of $A$ as a coherent sheaf on $S$, we get a 
short exact sequence 
\begin{equation}
  \label{eq:0}
  0 \hpil F(C,A) \hpil H^0 (A) \* _{\CC} \O_S  \hpil A \hpil 0
\end{equation}
of $\O_S$-modules, where $F(C,A)$ is locally free (since $A$ is locally 
isomorphic to $\O_C$ and hence has homological dimension one over $\O_S$).

Dualizing this sequence, one gets
\begin{equation}
  \label{eq:1}
  0 \hpil H^0 (A) \v \* \O_S \hpil E(C,A) \hpil 
    \N_{C/S} \* A \v \hpil 0,
\end{equation}
where $E(C,A) := {F(C,A)} \v$. This vector bundle has the 
following properties: 
\begin{eqnarray}
  \label{eq:2} & \det E(C,A) = L, \\ 
  \label{eq:3} & c_2 (E(C,A))= \deg A, \\ 
  \label{eq:4} & \rk E(C,A) =h^0(A), \\
  \label{eq:5} & h^i(E(C,A) \v)=  h^{2-i}(E(C,A) \* \omega_S)=0, \hs
  i=0,1, \\
  \label{eq:6} & h^0(E)=h^0(A)+h^0(\N_{C/S} \* A \v), \\
  \label{eq:7} & \mbox{If $A \leq \N_{C/S}$, then  $E(C,A)$ is 
               generated by its global sections off a finite set} \\
  \nonumber  & \mbox{coinciding with the (possibly zero) base divisor of 
$\N_{C/S} \* A \v$}. 
\end{eqnarray}



Note that if $A$ is any line bundle on $C$ computing the gonality or
the Clifford index of $C$,
then $A$ is generated by its global sections, and we can carry out the 
construction of the vector bundle $E(C,A)$ above.

Also recall that if $E$ is any vector bundle of rank $\geq 2$ 
on $S$ generated by its global sections and satisfying $c_1 (E) ^2>0$,
$h^0(E \v) = h^1(E \v)=0$, then $E= E(C,A)$ for some pair $(C,A)$,
where $C$ is a smooth curve in $|\det E|$ and $A$ is a non-trivial
line bundle on $C$ such that both $A$ and $\N_{C/S} \* A \v$ are base
point free. The conditions (\ref{eq:0})-(\ref{eq:7}) will be satisfied.
If $c_1 (E) ^2=0$, then $C$ is a union of non-intersecting smooth
curves all of self-intersection zero.

Similarly, if $V$ is a subspace of $H^0(A)$ spanning $A$ everywhere,
we can define a bundle $E(C,A,V)$ as the dual of a bundle $F(C,A,V)$
defined as above by\begin{equation}
  \label{eq:0'}
  0 \hpil F(C,A,V) \hpil V \* _{\CC} \O_S  \hpil A \hpil 0.
\end{equation}
and sitting in an exact sequence
\begin{equation}
  \label{eq:1'}
  0 \hpil V \v \* \O_S \hpil E(C,A,V) \hpil 
    \N_{C/S} \* A \v \hpil 0,
\end{equation}

The conditions (\ref{eq:2}), (\ref{eq:3}) and (\ref{eq:7}) are still
satisfied, but we get
\begin{equation}
  \label{eq:4'}
  \rk E(C,A) = \dim V \leq h^0(A),
\end{equation}
\begin{equation}
  \label{eq:5'}
  h^0(E(C,A) \v)=0, \hs h^1(E(C,A) \v)=h^0(A)-\dim V
\end{equation}
and
\begin{equation}
  \label{eq:6'}
  h^0(E)= \dim V+h^0(\N_{C/S} \* A \v) \leq h^0(A) +h^0(\N_{C/S} \* A \v).
\end{equation}

In the same manner we have that any globally generated vector bundle $E$ 
of rank $\geq 2$ satisfying $h^0(E \v)=0$ is of the form $E= E(C,A,V)$ for some
$V$ such that $h^0(A)-\dim V = h^1(E \v)$, and $A$ and $\N_{C/S}
\* A \v$ are both base point free.

Specializing to Del Pezzo surfaces, we get from (\ref{eq:1'}) tensored
by $\omega_S$ that
\begin{equation}
  \label{eq:8}
  h^1(A) = h^0(E (C,A,V) \* \omega_S).
\end{equation}
Moreover, we have $\N_{C/S} \* A \v \iso \omega_C \* A \v \*
\O_C(-K_S)$, and by (P1) we have
\begin{lemma} \label{quick}
  Assume $A$ and $\omega_C \* A \v$ are both base point free (as will
  be the case if $A$ computes the gonality or the Clifford index of
  $C$).
  
  Then $E(C,A)$ is generated by its global sections except possibly at the
  base point $x$ of $|-K_S|$, when $K_S^2=1$ and $C$ passes through $x$.
\end{lemma}




\section{Birational $k$-very ampleness and gonality of curves} \label{bir}

We first need the following easy result:

\begin{lemma} \label{low}
  Let $L$ be a nef line bundle on a Del Pezzo surface and $k \geq 1$ an 
  integer. Let $C$ be any smooth curve in $|L|$.

  If $g(L) \leq 2k$, then $(L+K_S)_C \iso \omega_C$ is not $k$-spanned.

  In particular, $C$ has gonality $\leq k+1$ and $L+K_S$ is not birationally 
  $k$-spanned. 
\end{lemma}

\begin{proof}
  By Theorem \ref{bs12}, $\omega_C$ is not $k$-spanned, and by Lemma \ref{gonkva}, all the smooth curves in $|L|$ have gonality $\leq k+1$. One easily sees that this implies the existence of a base point free linear system $A_C$ on each smooth $C \in L$ such that $h^0(A_C) \geq 2$ and $L+K_S$ fails to be $k$-very ample on any member of $|A_C|$. This implies that $L+K_S$ is not birationally $k$-spanned. 
\end{proof}

\begin{prop} \label{bfs}
  Let $k \geq 1$ be an integer and $L$ be a nef line bundle on a Del Pezzo surface $S$ such that
  $g(L) \geq \max \{2k-1,2 \}$. Let $\Z$
be any $0$-dimensional subscheme of $S$ such that $\deg \Z = k+1$ and such that the map
\[ H^0 (K_S \* L) \hpil H^0 (K_S \* L  \* \O _{\Z}) \]
is not onto, and for any proper subscheme $\Z'$ of $\Z$, the map 
\[ H^0 (K_S \* L) \hpil H^0 (K_S \* L  \* \O _{\Z'}) \]
is onto.

Then there are effective divisors $D$ and $M$ such that $L \sim D+M$,
$D$ passes through $\Z$, $M \geq -K_S$ and $D.M \leq k+1$. If  
$L^2 \geq 4k+3$, we furthermore have $M \geq D$.
\end{prop}

\begin{proof}
  This follows the same approach as \cite[Section 3]{kn}. For the sake of the 
  reader, we repeat the main arguments.

  From the first part of the proof of \cite[Thm. 2.1]{BS2} and from 
  \cite[(1.12)]{Ty}, it follows that there exists a rank 2 vector bundle $E$ on $S$ fitting
into the exact sequence 
\begin{equation} \label{BSes}
 0 \hpil \O_S \hpil E \hpil L \* \I _ {\Z} \hpil 0, 
\end{equation}
and such that the coboundary map of the exact sequence tensored with $K_S$,
\[ \delta : H^1 (L \* K_S \* \I _ {\Z}) \hpil H^2 (K_S) \iso H^0 (\O_S) \iso \CC, \]
is an isomorphism.

In particular, $h^1 (E \* K_S)=h^2 (E \* K_S)=0$.
  
Note that from the sequence (\ref{BSes}) we get $c_1 (E)^2 = L^2$ and 
$c_2 (E) = \deg \Z = k+1$. 

By assumption we have
\[ h^0(L \* K_S \* \I _ {\Z}) = h^0(L \* K_S)-k = g(L)-k \geq 1, \]
whence, by tensoring (\ref{BSes}) with $K_S$,
\begin{equation}
  \label{eq:ei}
 h^0(E \* K_S) >0. 
\end{equation}
This means that $E$ has a non-trivial line sub-bundle, and after saturating, we get an exact sequence
\begin{equation} \label{enes}
    0 \hpil M \hpil E \hpil N \* \I _ {A} \hpil 0,
  \end{equation}
where $M$ and $N$ are line bundles and $A \subset S$ is a zero-dimensional subscheme.

If $L^2 \geq 4k+3$, then by Riemann-Roch
\[ \chi (E \* E^{*} ) = c_1(E)^2 -4c_2(E)+4 =L^2 -4(k+1)+4 \geq 3, \]
whence $h^0 (E \* E^{*} ) \geq 2$ and $E$ is non-simple. 
  
One can then show (see \cite[Lemma 3.3]{kn}) that there are line bundles $M$ and $N$ and a zero-dimensional subscheme $A \subset S$ fitting into an exact sequence like in (\ref{enes}) satisfying either 
  \begin{itemize}
  \item [(a)] $M \geq N$, or
  \item [(b)] $A= \emptyset$ and the sequence splits.
  \end{itemize}

By symmetry we can assume $(M-N).K_S \geq 0$ in case (b). Hence we have
\begin{eqnarray*}
\chi (N-M) &  =   & \frac{1}{2}(N-M)(N-M-K_S) +1  \\
           &  =   & \frac{1}{2}(N-M)^2 - \frac{1}{2}(N-M).K_S+1 \\
           & \geq & \frac{1}{2}(L^2-4N.M)+1 \\
           & \geq & \frac{1}{2}(4k+3-4k-4) +1 >0, \\
\end{eqnarray*}
and by Riemann-Roch, we can assume $N \geq M$ in case (b) as well.

Exactly as in \cite[Lemma 3.5]{kn}, we can now prove that there exists a 
divisor $D \in |N|$ passing through $\Z$ and satisfying the numerical 
conditions above.
\end{proof}

If $L+K_S$ is not birationally $k$-very ample, there must exist a $\Z$ as 
above which is not contained in any divisor $Q$ with $h^0(Q)=1$, since
such divisors are necessarily supported on a union of $(-1)$-curves,
and there are finitely many such curves on a Del Pezzo surface.
Therefore $h^0(D) \geq 2$ and since $M \geq -K_S$, we must have $h^0(M) \geq 2$ also.

Furthermore, assuming $L+K_S$ is birationally $(k-1)$-very ample, we
have by Lemma \ref{low} that $g(L) \geq 2k-1$.
This means that we have the following:

\begin{prop} \label{cond}
Let $k \geq 1$ be an integer and $L$ a nef line bundle of sectional
genus $g(L) \geq 2$ on a Del Pezzo surface. If $L+K_S$ is not
birationally $k$-very ample, there exists an effective divisor $D$ satisfying the following conditions:
\begin{eqnarray*}
 (\dagger)   & h^0(D) \geq 2,  \hs h^0(L-D) \geq 2, \hs
  D.(L-D) \leq k+1, \\
  & L+K_S \geq D, \hs L \geq 2D \hs \mbox{ if } \hs L^2 \geq 4k+3.
\end{eqnarray*}
\end{prop}


\begin{rem} \label{newself}
  {\rm (i) As a consequence of the properties $(\dagger)$ above, we
also have, when \linebreak $L^2\geq 4k+3$, 
\begin{eqnarray}
  \label{eq:ex1} D^2 \leq (L-D)^2 \hs \mbox{ and } \\
  \label{eq:ex2} D^2 \leq k+1, \hs \mbox{ with equality only if } \hs
                 L \sim 2D.
\end{eqnarray}
These properties will often be used without further mention.

  (ii) If $g(L) \geq 2k+1$, we have $L^2 \geq 4k+3$. Indeed, by the adjunction formula, $L^2 =2g(L)-2-L.K_S \geq 4k-L.K_S \geq 4k+2$ by (P6). If equality holds, then $-L.K_S=2$, and by the Hodge index theorem and the fact that $g(L) \geq 2k+1 \geq 3$, this is impossible.

  (iii) The condition $L+K_S \geq D$ is automatically satisfied when $L \geq 2D$. Indeed, if $M :=L-D \not \geq -K_S$, then also $D \not \geq -K_S$, whence $D.(D+K_S) \leq -2$ and $M.(M+K_S) \leq -2$. By Lemma \ref{low}, we can assume $L.(L+K_S) \geq 4k-4$, whence $4k-4 \leq L.(L+K_S) = M.(M+K_S)+D.(D+K_S) +2M.D \leq 2k-2$, and we get $k=1$ and $L.(L+K_S)=0$, contradicting the fact that $g(L) \geq 2$. }
\end{rem}



We also have a similar result concerning the gonality of a smooth
curve in $|L|$:

\begin{prop} \label{firstgon}
  Let $k \geq 1$ be an integer and $L$ a nef line bundle of sectional
  genus $g(L) \geq 2$ on a Del Pezzo surface. Assume there is a smooth
  curve $C \in |L|$ of gonality $k+1$, and let $A$ be a $g^1_{k+1}$ on
  $C$. Then there exists an effective divisor $D$ satisfying the
  conditions  $(\dagger)$ and such that $\O_C(D) \geq A$. 
\end{prop}

\begin{proof}
If $\Z \sub C$ is any  $0$-dimensional subscheme, then we have a 
commutative diagram with exact rows and columns 
\begin{equation}  \label{diagram}
\xymatrix{
           & 0 \ar[d] & 0 \ar[d]  & & 
\\
           & K_S  \ar@{=}[r] \ar[d] & K_S\ar[d] & & 
\\
  0 \ar[r] & (L+K_S) \* {\I}_{\Z} \ar[r] \ar[d] & L+K_S \ar[r]^{\alpha_{\Z}} \ar[d] & 
    (L+K_S)\* {\O}_{\Z} \ar[r] \ar@{=}[d] & 0
\\ 
  0 \ar[r] & \omega_C (-\Z) \ar[r] \ar[d] & \omega_C \ar[r]^{\beta_{\Z}}  \ar[d] & 
   \omega_C \* {\O}_{\Z} \ar[r] & 0 
  \\
  & 0 & 0.  & &
} 
\end{equation}
From  the fact that $h^1(L+K_S)=0$ and $h^1(\omega_C)=1$, we see that the following conditions are equivalent:
\begin{itemize}
  \item[(i)]   $H^0(\alpha_{\Z})$ is not surjective,
  \item[(ii)]  $H^0(\beta_{\Z})$ is not surjective,
  \item[(iii)]  $h^0 (\O_C(\Z)) \geq 2$.
\end{itemize}

Now let $\Z$ be any element in $|A|$, which we can choose to lie
outside of the finitely many $(-1)$-curves on the surface, since $A$
is base point free. Since $C$ is $(k+1)$-gonal, we must have $h^0
(\O_C(\Z'))=1$ for any proper subscheme $\Z'$ of $\Z$. Then the
assumptions in Proposition \ref{bfs} are satisfied, and we can find a
divisor $D$ passing through $\Z$ with $h^0(D) \geq 2$ and satisfying
the conditions $(\dagger)$. Tensoring the left hand vertical exact
sequence above with $D-L-K_S$ and taking cohomology, we get
\[ h^0(\O_C(D)(-A)) \geq h^0(D \* {\I}_{\Z}) >0. \]   
\end{proof}

\begin{example} \label{only}
  {\rm Let $L \sim -2K_{S_8}$. Then one easily computes $g(L)=2$, so all
  the smooth curves in $|L|$ are hyperelliptic. Moreover, the divisor
  $D$ given by Proposition \ref{firstgon} is $-K_{S_8}$, and the $g^1_2$ on any
  such $C \in |L|$ is given by $\O_C(-K_{S_8})$.

  Furthermore $L+K_{S_8}$ fails to be $k$-spanned on any zero scheme of
  these $g^1_2$'s, so $L+K_{S_8}$ is not birationally $k$-very ample.

  Also note that $D.(L-D)= K_{S_8}^2=1 < k+1=2$ in this case.} 
\end{example}

Among all the divisors satisfying the conditions $(\dagger)$, the ones satisfying these conditions for the smallest value of $k$ play an important role. The next result is similar to \cite[Prop. 5.1]{kn} for $K3$ surfaces. We recall the definition of $\R(L)$ in the introduction. We will also need to define the set
\[ \R_1 (L) := \{ \Gamma \hs | \hs \Gamma=(-1)\mbox{-curve} \hs \mbox{and} \hs\Gamma.L = 1 \}. \]

\begin{prop} \label{findD}
Assume $L$ is  a big and nef line bundle of sectional genus $g(L) \geq
2$ on a Del Pezzo surface $S$ such that $L \not \sim -2K_{S_8}$
Assume there are divisors satisfying the conditions $(\dagger)$ for
$k=k_0$, but none for $k < k_0$. Then $k_0 \geq 1$, and any divisor $D$ satisfying $(\dagger)$ for $k=k_0$ has the following properties (with $M := L-D$):
\begin{itemize}
\item[(i)]   $D.M = k_0 +1$;
\item[(ii)]  $h^1 (D)=h^1 (M)= h^1 (D+K_S)=h^1 (M+K_S)=0$; 
\item[(iii)] $M$ and $D$ are either base point free or $\sim
  -K_{S_8}$. Moreover $g(M) \geq 1$;
\item[(iv)]  the general members of $|M|$ and $|D|$ are smooth curves.
\end{itemize}
Furthermore $D$ is of one of the following types (with $g(D)=0$ in
(a) and (b) and $g(D)=1$ in (c)-(g)):
\begin{itemize}
\item[(a)] $D^2=0$, $D.K_S=-2$,
\item[(b)] $D^2=1$, $D.K_S=-3$,
\item[(c)] $D \sim -K_{S_8}$,
\item[(d)] $D^2=-K_S.D=2$; \hs $K_S^2 \leq 2$ with $D \sim -K_{S_7}$ if $K_S^2=2$,
\item[(e)] $D^2=-D.K_S=3$; \hs $K_S^2 \leq 3$ with $D \sim -K_{S_6}$ if $K_S^2=3$,
\item[(f)] $D^2=-D.K_S=4$, $g(L)=5$, $k_0=3$; \hs  $K_S^2 \leq 4$ with $D \sim -K_{S_5}$ if $K_S^2=4$. 
\end{itemize}
Furthermore, if there is a divisor as in (b) for 
$(k_0,g(L)) \not = (2,3)$ or as in (e) for \linebreak 
$(k_0,g(L)) \not = (2,4),(3,5)$, then all the
smooth curves in $|L|$ have Clifford index $k_0-2$ and gonality
$k_0+1$, whence are exceptional.
\end{prop}

\begin{proof}
   Note that by Proposition \ref{firstgon} and Lemma \ref{low}, we
   have
   \begin{eqnarray}
     \label{eq:sil}  & \mbox{   All the smooth curves in $|L|$ have
   gonality $\geq k_0+1$,} \hs \mbox{ and } \\
     \label{eq:ke}   & g(L) \geq 2k_0-1.
   \end{eqnarray}

Condition (i) is immediate (see \cite[Section 5]{kn}).

  We first want to show that $D$ is nef (the same reasoning will work for $M$). If $D$ is not nef, there exists a $(-1)$-curve $\Gamma$ such that $\Gamma.D <0$. Let $D' := D- \Gamma$ and $M' := M+ \Gamma$. Then
\begin{eqnarray}
\nonumber  D'.L -{D'}^2 = D'.M' & = & (D-\Gamma).(M+ \Gamma) = 
                            D.M - \Gamma.M +  \Gamma.D +1 \\
\label{nefness}  & = & D.M + 2\Gamma.D - \Gamma.L +1 \\
 \nonumber   & < & D.M = k_0+1,
\end{eqnarray}
a contradiction. 

So $D$ and $M$ are nef, whence by property (P4) they are either base
point free or linearly equivalent to $-K_{S_8}$.
By (P5), we have $h^i(M)=h^i(D)=0$, for $i=1,2$.

Now we show that $k_0 \geq 1$. So assume to get a contradiction that
$D.M=1$. Assume first that $M^2 \geq
D^2$. Then by the Hodge index theorem we must have $D^2 =0$ or $1$, the latter case implying $D \sim M$. If $D^2 =0$, then $D.K_S \leq -2$
and $D.(L+K_S) = D^2+D.M +D.K_S \leq -1$, contradicting the nefness of
$D$ (note that $L + K_S >0$ since $g(L) \geq 2$). If $D^2=1$, then we
must have $D.K_S \leq -3$, since otherwise $D \sim -K_{S_8}$ and 
$L \sim -2K_{S_8}$. Again we get the contradiction $D.(L+K_S) = D^2+D.M +D.K_S \leq -1$. 

The same reasoning works if $M ^2 \leq D^2$. This shows that $k_0 \geq 1$.

Next we prove that $h^1(D+K_S)=0$ and that the generic member of $|D|$ is a smooth curve. The same reasoning will work for $M$.

If $D^2 >0$, then $D$ is big and nef and $h^1(D+K_S)=0$. This implies $h^0(\O _{D_0})=1$ for all $D_0 \in |D|$. Hence by Bertini, the general member of $|D|$ is a smooth curve. 

If $D^2=0$ and $h^1(D+K_S) \not =0$, we have $h^0(\O _{D_0}) > 1$ for all $D_0 \in |D|$, whence $D$ is not numerically $1$-connected. This means that there is an effective decomposition
\[ D \sim D_1 + D_2, \]
such that $D_1.D_2 \leq 0$. Since $D$ is nef, we must have 
$D.D_1=D.D_2=D_1.D_2=D_1^2=D_2^2=0$. One then easily sees that both $D_1$ and 
$D_2$ will satisfy the conditions $(\dagger)$ for some integer $k < k_0$, a 
contradiction. Whence $h^1(D+K_S) =0$, $h^0(\O _{D_0})=1$ and the general 
member of $|D|$ is a smooth curve. 

Since $M \geq -K_S$ and $h^1(M+K_S)=0$, we have $h^0(M+K_S)=g(M) \geq 1$.

We will now prove that $D$ is of one of the types (a)-(f), and that the last statement in the proposition is satisfied.

We first show that $g(D)=0$ or $1$.

By Riemann-Roch and the fact that $h^1(D+K_S)=0$, we have
\begin{equation} \label{eq:65}
h^0(D+K_S) = g(D).  
\end{equation}

If $-D.K_S \leq -M.K_S$, we have

\begin{equation} \label{eq:65b}
(D+K_S).(M-K_S) =D.M + K_S.(M-D) -K_S^2 < D.M,
\end{equation}
which yields $h^0(D+K_S)=g(D) \leq 1$ (since clearly $h^0(M-K_S) \geq 2$).

If $-D.K_S > -M.K_S$, we have by (\ref{eq:65}) with 
$M$ and $D$ interchanged, that $g(M) \leq 1$, whence $g(M)=1$.
Furthermore, by $(\dagger)$, Remark \ref{newself}(ii) and (\ref{eq:ke}) we have $g(L) =2k_0-1$ or $2k_0$, and $L^2 \leq 4k_0+2$. By the adjunction formula, we have $-L.K_S \leq 6$ and $4$ as  $g(L) =2k_0-1$ or $2k_0$, respectively.

We have 
\begin{equation}
  \label{eq:and}
  L.(L+K_S) = \left\{ \begin{array}{ll}
             4k_0-4 & \mbox{ if } \hs g(L) =2k_0-1, \\
             4k_0-2 & \mbox{ if } \hs g(L) =2k_0. 
          \end{array}
    \right . 
\end{equation}
Since $L.(L+K_S) = M.(M+K_S) + D.(D+K_S)+2M.D= D.(D+K_S)+2k_0+2$, we get
\begin{equation}
  \label{eq:reas}
  D.(D+K_S) = \left\{ \begin{array}{ll}
             2k_0-6 & \mbox{ if } \hs g(L) =2k_0-1, \\
             2k_0-4 & \mbox{ if } \hs g(L) =2k_0. 
          \end{array}
    \right . 
\end{equation}

We have $k_0+1 \leq (-K_S).(L+K_S) = -K_S.L -K_S^2$, whence $k_0 \leq 4$, with equality only if $K_S^2=1$. Therefore, if $g(D) \geq 2$, by (\ref{eq:reas}) we must have $k_0 = 4$, $g(D)=2$. $-L.K_S=6$ and $S \iso S_8$. In this case, all the smooth curves in $|D|$ are hyperelliptic, whence by Proposition \ref{firstgon}, we have an effective decomposition
\[ D \sim A+B, \]
with $h^0(A) \geq 2$ and $h^0(B) \geq 2$ and such that $A.B \leq 2$. We can assume $A.M \geq B.M$. We get
\[ M.D \leq M.(A+B) \leq M.D +2 -A.M, \]
whence $A.M \leq 2$ and $k_0+1 =D.M \leq 4$, whence the contradiction $k_0 \leq 3$.

So we have proved that $g(D) \leq 1$.

Now let $g(D) =0$. 
Let $C$ be a smooth curve in $|L|$. By the exact sequence
\begin{equation} \label{eq:62}
  0 \hpil -M \hpil D  \hpil D_C \hpil 0
\end{equation}
and the fact that $h^1(-M)=0$, we get $h^0(D_C)=h^0(D)=D^2+2$, whence
\begin{equation}
  \label{eq:63}
  \Cliff D_C=D.L-2(h^0(D)-1)=D.L-2D^2-2 = k_0-D^2-1.
\end{equation}

By (\ref{eq:62}) we have $h^1(D_C) =h^0(M+K_S) \geq 1$.

By (\ref{eq:007}) and (\ref{eq:ke}) we have
\begin{eqnarray}
  \nonumber h^1(D_C) & =    & g(L)+1 - \Cliff D_C - h^0(D_C) \\
  \label{eq:lek}     & \geq & 2k_0-(k_0-D^2-1)-(D^2+2)       \\
  \nonumber          & =    & k_0-1.
\end{eqnarray}



Assume that $D^2 \geq 2$. By (\ref{eq:lek}) we have $\Cliff D_C \leq k_0-3$.
If $h^1(D_C) \geq 2$, then $D_C$ contributes to the Clifford index of
 $C$ and by Lemma \ref{gon1} we get the contradiction
 \begin{equation}
   \label{eq:64}
   \gon C \leq \Cliff C +3 \leq \Cliff D_C +3   =k_0-3+3=k_0.
 \end{equation}
 
So we must have $h^1(D_C) =1$, whence $k_0 \leq 2$ and $\Cliff D_C \leq -1$, contradicting Clifford's theorem.

So  $D^2 \leq 1$ when $g(D)=0$,
which gives us the cases (a) and (b).

Assume now $D^2=1$, which means that we are in case (b). Then by
(\ref{eq:63}) we have $\Cliff D_C=k_0-2$. If $h^1(D_C) \geq 2$, then all the smooth curves in $|L|$ are exceptional (of Clifford index $k_0-2$ and gonality $k_0+1$). If $h^1(D_C) = 1$, then by (\ref{eq:lek}) and Clifford's theorem, we have $k_0 =2$ and $g(L) = 2k_0-1=3$. 



So if we are in case (b) with $(k_0,g(L)) \not = (2,3)$, then all the smooth curves in $|L|$ have
Clifford index $k_0-2$ and gonality $k_0+1$, whence are exceptional (of
Clifford dimension $h^0(D_C)=h^0(D)=3$).

Let $g(D) =1$. By (\ref{eq:62}) again, we get $h^0(D_C)=h^0(D)=D^2+1$, whence
\begin{equation}
  \label{eq:67}
  \Cliff D_C=D.L-2(h^0(D)-1)=D.L-2D^2 = k_0+1-D^2.
\end{equation}
Also,
\begin{eqnarray}
  \nonumber h^1(D_C) & =    & g(L)+1 - \Cliff D_C - h^0(D_C) \\
  \label{eq:68}     & \geq & 2k_0-(k_0-D^2+1)-(D^2+1)       \\
  \nonumber          & =    & k_0-2.
\end{eqnarray}

Assume now $D^2 \geq 4$. Then we have $\Cliff D_C \leq k_0-3$. If $h^1(D_C) \geq 2$, we get a contradiction as in (\ref{eq:64}),
whence $k_0 = 3$ by Clifford's theorem and $g(L)= 2k_0-1=5$ by (\ref{eq:68}).

This means that $D^2 \leq 4$ when $g(D)=1$, with equality implying
$k_0=3$ and $g(L)=5$.

We therefore get the cases (c)-(f) above. By (P6) $D \sim -K_{S_8}$ if $D^2=-D.K_S=1$.

Assume now $D$ is as in (e), i.e. $D^2=3$ and $g(D)=1$. Then by
(\ref{eq:67}) we have $\Cliff D_C =k_0-2$. If $h^1(D_C) \geq 2$, then all the smooth curves in $|L|$ are exceptional (of Clifford index $k_0-2$ and gonality $k_0+1$). If $h^1(D_C) =1$, then $k_0 \leq 3$ by (\ref{eq:68}) and $k_0 \geq 2$ by Clifford's theorem. By (\ref{eq:68}) we get the two cases $(k_0,g(L))=(3,5)$ and $(2,4)$.
  
So if we are in case (g), and $(k_0,g(L)) \not = (2,4), (3,5)$, then all the smooth curves in $|L|$ have
Clifford index $k_0-2$ and gonality $k_0+1$, whence are exceptional (of
Clifford dimension $ \leq h^0(D_C)-1=h^0(D)-1=3$).
\end{proof}


\begin{rem} \label{redundant}
  {\rm In the cases with $g(D)=1$, i.e. the cases (c)-(f), of the proposition, one can say more about the divisors $D$. In fact we have $h^0(D+K_S)=1$, so 
\[ D + K_S\sim A,\]
for some divisor $A \geq 0$ satisfying 
$h^0(A) = \frac{1}{2}A.(A-K_S) +1 =1$, whence $A^2 = A.K_S$.
Furthermore, $A$ is either trivial or only supported on a 
union of disjoint $(-1)$-curves. We get
\begin{eqnarray*}
  D.L-D^2  & = & (-K_S + A).L - (-K_S + A)^2 \\
           & = & -K_S.L- K_S^2 +A.L +2A.K_S - A^2 \\
           & = & -K_S.L- K_S^2 +A.(L + K_S). 
\end{eqnarray*}
By assumption we must have $D.L-D^2 \leq -K_S.L- K_S^2$, so we easily see that $A$ has to be supported on $\R(L)$ and $\R_1(L)$, so that we can write 
$D \sim -K_S + \sum_{\Gamma_i \in \R(L)} n_i \Gamma_i 
    + \sum_{\Gamma'_i \in \R_1(L)} m_i \Gamma'_i$.  
To achieve the minimum it is a quick exercise to find that all $n_i=1$
and $m_i=0$ or $1$ (using the fact that all the curves in $\R(L)$ and
$\R_1(L)$ are disjoint), so that we have
\begin{equation}
  \label{eq:form}
D \sim -K_S + \sum_{\Gamma_i \in \R(L)} \Gamma_i 
    + \sum_{\Gamma'_i \in \B} \Gamma'_i,  
\end{equation}
for a subset $\B$ of $\R_1(L)$.

At the same time one also sees that the divisor 
\begin{equation}
  \label{eq:formII}
D \sim -K_S + \sum_{\Gamma_i \in \R(L)} n_i \Gamma_i  
\end{equation}
also satisfies the conditions $(\dagger)$ for $k=k_0$.

One then easily sees that case (c) implies that $L$ is ample, and that there are three possibilities for case (d):
\begin{itemize}
\item[(d1)] $D \sim - K_{S_7}$ (only if $L$ is ample),
\item[(d2)] $D \sim -K_{S_8}+ \Gamma$, with $\R(L) = \{ \Gamma\}$ 
           and $\R_1(L) = \emptyset$,
\item[(d3)]  $D \sim -K_{S_8}+ \Gamma$, with $\R(L) = \emptyset$ 
           and $\R_1(L) = \{ \Gamma\}$.
\end{itemize}

The corresponding
possibilities for the cases (g) and (f) are left to the reader.

We will see below that if $C$ is exceptional, then there exists
  a divisor $D$ as in (b), or as in (c) with $g(L) =10$ and $c=3$. Moreover, $D_C$ will compute the Clifford
  dimension of $C$. 
}
\end{rem}






















































With the help of Proposition \ref{findD}, we now show 

\begin{prop} \label{fail}
  Let $L$ be a big and nef line bundle on a Del Pezzo surface with
  $g(L) \geq 2$. If there is a divisor satisfying the conditions 
  $(\dagger)$, then $L+K_S$ is not birationally $k$-spanned and there
  is a smooth curve in $|L|$ of gonality $\leq k+1$.
\end{prop} 

\begin{proof}
  We have seen that the result holds for $L \sim -2K_{S_8}$ in Example
  \ref{only}. So we can assume $L \not \sim -2K_{S_8}$.
  By Proposition \ref{findD} we can assume $D.L=D^2+k+1$
  and that the general member of 
  $|D|$ is a smooth curve. Define $M:=L-D$. 

 Assume first $M,D \not \sim -K_{S_8}$. Then by 
Proposition
  \ref{findD}(iii), both $M$ and $D$ are base point free. 
  As is the proof of
  \cite[Prop. 4.3]{kn}, $L+K_S$ fails to 
  be $k$-spanned on any section of $M_{D'}$, for any smooth $D' \in |D|$. 
  Furthermore, one finds $h^0(M_{D'}) \geq 2$ and $|M_{D'}|$ is base point 
  free since $M$ is. Since any 
  Zariski-closed proper subset of $S$ will contain at most finitely many 
  of these curves and 
  intersect the rest of them in a finite number of points, the assertion  
  follows.

Also, the rank two bundle $E := M \+ D$ is  base point free, and $h^1(D)=h^1(M)=0$, $\det E =L$, $c_2(E)=M.D=k+1$, so $E=E(C,A)$ for some smooth curve $C \in |L|$ and some non-trivial base point free line bundle $A$ on $C$ satisfying $h^0(A) = 2$ and $\deg A=c_2(E)=k+1$. Hence $C$ has gonality $\leq k+1$.

Finally, assume $D \sim -K_{S_8}$. Then $D.L =k+2$, and any smooth curve in $|L|$ passing through the base point $x$ of $|-K_S|$ clearly has a $g^1_{k+1}$. The same holds if $M \sim -K_{S_8}$. Since these curves form a positive dimensional family, and each such curve has a base point free linear system $|A|$ such that $L+K_S$ fails to be $k$-very ample on any member of $|A|$, it follows that $L+K_S$ is not birationally $k$-very ample.
\end{proof}

We will now give the proof of Theorem \ref{birkvathm}.

\renewcommand{\proofname}{Proof of Theorem \ref{birkvathm}}

\begin{proof}
From Propositions \ref{bfs}, \ref{firstgon} and \ref{fail} it is clear that (a), (b) and (d) are equivalent conditions.

Furthermore, by Lemma \ref{low}, we must have $g(L) \geq 2k+1$, if any of these conditions are satisfied, whence $L^2 \geq 4k+3$ by Remark \ref{newself}.

Now by Propositions \ref{bfs}, \ref{findD} and \ref{fail}, (a) is equivalent to the condition that there exists no divisor $D$ satisfying
\begin{eqnarray*}
  &  D.L-D^2=k+1 \hs \mbox{ for } \\  & g(D)=0 \hs \mbox{ and } \hs D^2=0,1, \hs  \mbox{ or }  \hs  g(D)=1 \hs \mbox{ and } \hs 1 \leq D^2 \leq \min \{k+1,4 \}.
\end{eqnarray*}
Such a divisor with $g(D)=0$ and $D^2=0$ gives $D.L=k+1$ and with $g(D)=0$ and $D^2=1$ gives $D.L=k+2$.

Moreover, by Remark \ref{redundant} we can assume that such a divisor with $g(D)=1$ is of the form
\[ D \sim -K_S + \sum_{\Gamma \in \R(L)} \Gamma, \]
for $D^2 =K_S^2+m(L) \leq \min \{k+1,4 \}$, which gives
\[ -K_S.L =D.L = D^2 +k+1 =K_S^2+m(L)+k+1. \]
This shows the equivalence of (a), (b) and (d) with the numerical conditions in (c).

Now assume that $L+K_S$ is base point free, so that it defines a morphism
\[ \phi_{L+K_S}: S \hpil \PP^{g(L)-1}. \]
We will prove that the condition (e) is then equivalent to (a).

Let $Z$ be a zero-dimensional subscheme of $S$ of length $k+1$. It is well-known that the map
\[ H^0(L+K_S) \hpil H^0((L+K_S) \* \O_Z) \]
is not surjective if and only if $\phi_{L+K_S}(S)$ has a $(k+1)$-secant $(k-1)$-plane through the image of $Z$.

Assume now $L+K_S$ is birationally $k$-very ample. In particular $g(L) \geq 2k+1$ by Lemma \ref{low}. 

If a $Z$ as above exists, then by Proposition \ref{bfs}, there is an effective divisor $D$ passing through $Z$ and such that $D.(L-D) \leq k+1$. Since $L+K_S$ is birationally $k$-very ample, $|D|$ cannot move by Proposition \ref{fail}, whence $h^0(D)=1$ and $D$ is supported on a union of $(-1)$-curves. This precisely means that the $(k+1)$-secant $(k-1)$-plane through the image of $Z$ contains the images of these $(-1)$-curves.

Conversely, if all the $(k+1)$-secant $(k-1)$-planes of $\phi_{L+K_S}(S)$ contain images of $(-1)$-curves of degree $\leq k$, $L+K_S$ is clearly $k$-very ample outside the finitely many $(-1)$-curves of $S$, whence birationally $k$-very ample.
\end{proof}

\renewcommand{\proofname}{Proof}


As an easy application, we prove the following result, which will be needed later.

\begin{prop} \label{easyapp}
  Let $L$ be a base point free line bundle on a Del Pezzo surface $S$ and denote by $\phi_L$ the morphism defined by
  $L$.
  
  Then the following conditions are equivalent:
  \begin{itemize}
  \item[(i)] $\phi_L$ is birational,
  \item[(ii)] $\phi_L$ is an isomorphism
  outside of finitely many contracted $(-1)$-curves,
  \item[(iii)] $L^2>0$, $(L^2,-K_{S_8}.L) \not = (2,2)$, $L \not \sim
    -2K_{S_8}$ and $L \not \sim -K_{S_7}$,
  \item[(iv)] none of the smooth curves in $|L-K_S|$ are hyperelliptic.
  \end{itemize}
\end{prop}

\begin{proof}
  By Theorem \ref{birkvathm}, the conditions (i), (ii) and (iv) are equivalent. We have to prove that the numerical conditions in (iii) are equivalnet to the ones in Theorem \ref{birkvathm}(c).

Note that since $L$ is base point free, $L-K_S$ must be ample, whence $m(L-K_S)=0$. The negations of the conditions in Theorem \ref{birkvathm}(c) are then: There exists a smooth rational curve $D$ satisfying $(D^2, L.D)=(0,0)$ or 
$(1,0)$, or $-K_S.L \leq 2$. In the two first cases, the Hodge index theorem yields $L^2=0$, $L \sim D$. In the last case, either $L^2=0$ again or $(L^2,-K_{S_8}.L) = (2,2)$, $L \sim
    -2K_{S_8}$ or $L \sim -K_{S_7}$ (since $L \not \sim -K_{S_8}$).
\end{proof}

We also state the result for $k=2$, leaving the proof to the reader:

\begin{prop} \label{easyappII}
  Let $L$ be a base point free line bundle on a Del Pezzo surface $S$ and denote by $\phi_L$ the morphism defined by $L$.
  
  Then the following conditions are equivalent:
  \begin{itemize}
  \item[(i)] $L$ is birationally $2$-very ample,
  \item[(ii)] $\phi_L$ is birational and $\phi_L (S)$ has only finitely many trigonal lines (outside of the possible rational double points), and these are all lines contained in the surface, 
  \item[(iii)] $L^2>1$, $-K_S.L >3$ and $L.D \geq 2$ for any smooth rational curve $D$ with $D^2=0$,
  \item[(iv)] none of the smooth curves in $|L-K_S|$ are hyperelliptic or trigonal.
  \end{itemize}
\end{prop}


\section{The constancy of the gonality of smooth curves in $|L|$} 
\label{constgon}

Given that the minimal gonality of a smooth curve in $|L|$ is $k+1$, for an 
integer $k \geq 1$ (which by Theorem \ref{birkvathm} is equivalent to
$L+K_S$ being birationally $(k-1)$-very ample but not birationally
$k$-very ample), we will in this section study the gonality of all the
smooth curves in $|L|$. For $\deg S \geq 2$, this will coincide with the results of Pareschi \cite{par}, but for the sake of completeness, we do not restrict to $\deg S=1$.


\begin{prop} \label{constgonprop}
  Let $L$ be a nef line bundle of sectional genus $g(L) \geq 2$ on a
  Del Pezzo surface such that $L \not \sim -2K_{S_8}$. Assume that
  the minimal gonality of a smooth curve in $|L|$ is $k+1$, for an 
  integer $k \geq 1$ and that among all the divisors satisfying
  $(\dagger)$, there is one not being of one of the following types:
\begin{itemize}
  \item[(a)] $k=1$, $D \sim -K_{S_7}$ and $L \sim -2K_{S_7}$,
  \item[(b)] $k=1$, $D \sim -K_{S_8}+ \Gamma$, with 
             $\Gamma$ a $(-1)$-curve and $L \sim 2D$,
  \item[(c)] $D \sim -K_{S_8}$.
\end{itemize}
Then all the smooth curves in $|L|$ have gonality $k+1$.
\end{prop}

\begin{proof}
  If there exists a $D$ satisfying $(\dagger)$ with $g(D)=0$, we have 
  seen before that $h^0(D_C)=h^0(D)= D^2+2$
and $\deg D_C = D^2+k+1$ for any $C \in |L|$.

If $D^2=0$, then $D_C$ is a $g^1_{k+1}$, and if $D^2=1$, then for any
$x \in C$ we have
$h^0(D_C-x) \geq 2$ and $\deg(D_C-x)  =k+1$, whence $C$ has a $g^1_{k+1}$.

  If $D$ satisfies $(\dagger)$ with $g(D)=1$, we have $h^0(D_C)=h^0(D)= D^2+1$,
and $\deg D_C = D^2+k+1$. As seen in Proposition \ref{findD} we have
$D^2 \leq 4$. Let $n:=D^2-1$. If $D_C$ is not $n$-very ample, there exists a divisor $A$ on $C$ of degree $n+1=D^2$ such that 
$h^0(D_C-A)  \geq h^0(D_C) -n =2$. Since $\deg(D_C-A)  =k+1$, we have that $C$ has a $g^1_{k+1}$.

So assume $D_C$ is $n$-very ample for some smooth $C \in |L|$. By (\ref{eq:62})
we see that $h^1(D_C) = h^2(-M)=h^0(M+K_S) \geq 1$. By \cite[Thm. 2.9]{ba}, we must have $n+2=h^0(D_C) \geq 2n+1$, which is only possible if $n=0$ or $1$, i.e. $D^2=-D.K_S=1$ or $2$. By the Hodge index theorem, this leaves us with the three possibilities
\begin{itemize}
  \item[(i)]   $\deg S=2$, $D \sim -K_S$,
  \item[(ii)]  $\deg S=1$, $D^2=-D.K_S=2$, 
  \item[(iii)] $\deg S=1$, $D \sim -K_S$.
\end{itemize}

If we are in case (i) and the gonality is not constant, it means that there exists a smooth curve $C \in |L|$ such that $D_C$ is very ample, i.e. $C$ is isomorphic to a smooth plane curve of degree $D.C= k+3$. Using the formula for the genus of a smooth plane curve, we calculate the self-intersection of $C$:
\[ C^2 =2g(C)-2+C.D =(k+1)(k+2) + (k+1) = (k+1)(k+3), \]
whence by the Hodge index theorem
\[ 2(k+1)(k+3) = C^2D^2 \leq (C.D)^2 =(k+3)^2, \]
which gives $k=1$ and $L \sim 2D$. This gives case (a) above.

If we are in case (ii), then since $h^0(D+K_S)=1$ and $D \not \sim -K_S$, we have $D \sim -K_S + \Gamma$, where $\Gamma$ is supported on a union of $(-1)$-curves. Since $-D.K_S=D^2=2$, we get $-K_S.\Gamma=1$, whence $\Gamma$ is a smooth $(-1)$-curve. From 
\[ (D-\Gamma).L - (D-\Gamma)^2=(D-\Gamma).(M+\Gamma) = D.M-\Gamma.M+\Gamma.D+1=D.M-\Gamma.L+1, \]
we must have $\Gamma.L \leq 1$. 

As in case (i), if the gonality is not constant, there has to exist a smooth curve $C \in |L|$ which is isomorphic to a smooth plane curve of degree $D.C= k+3$.
Again we calculate the self-intersection of $C$:
\[ C^2 =(k+1)(k+3)-\Gamma.L, \]
and the Hodge index theorem gives $k=1$ and the two possibilities $L \sim 2D$ 
for $\Gamma.L=0$ and $L^2=7$ for $\Gamma.L=1$. In the latter case, we get the 
contradiction $2D.L =8 > L^2$ (since $L^2 \geq 4k+3$). So we must be in the first case, which
is case (b) above.

This concludes the proof of the proposition.
\end{proof}

Case (a) in the proposition is exactly Case (I) mentioned in the introduction, the example already treated by Pareschi, where there is a codimension $1$ family of smooth hyperelliptic curves in 
$|L|$, whereas the general smooth curve is trigonal. 

Case (b) is exactly Case (II) mentioned in the introduction. Since
$-K_S$ satisfies the conditions $(\dagger)$ for $k=1$ with respect to
$D-K_S$, we have that $D$ is not birationally very ample by
Proposition \ref {fail}, whence $|D|$ gives a morphism
$\phi: S \khpil \PP^2$, which is not birational, and then easily seen
to be $2:1$ and ramified along a smooth quartic by the Riemann-Hurwitz formula. This case is then
analogous to Case (I). 

From the proof of Proposition \ref{constgonprop} we have the following
description of the codimension $1$ family of smooth hyperelliptic curves:

\begin{prop}
  The codimension $1$ family of smooth hyperelliptic curves in $|L|$
  in Case (I) (resp. Case (II)) is exactly the family of smooth curves
  passing through the $0$-dimensional schemes of length two where
  $-K_S$ (resp. $D$) fails to be very ample.
\end{prop}

We will now show that if the only divisor satisfying $(\dagger)$ for $k=k_0$ 
is $-K_S$, then the smooth curves in the codimension one 
subfamily of $|L|$ consisting of smooth curves passing through the base point 
of $|-K_S|$ all have gonality $k_0+1$, whereas the smooth curves outside this family have gonality $k_0+2$.

Denote by $x$ the base point of $|-K_S|$. As we have seen in the last part of 
the proof of Proposition \ref{fail}, any smooth curve in $|L \* \I_x|$ has a $g^1_{k_0+1}$, whence $C$ is $(k_0+1)$-gonal. Moreover, since $L$ is base point free, $\dim |L \* \I_x| = \dim |L|-1$, whence there is a codimension one family of such curves.

Assume now $C \in |L|$ is smooth of gonality $k_0+1$. We will show that $C \in 
|L \* \I_x|$. 

Let $|A|$ be a (necessarily complete and base point free) $g^1_{k_0+1}$ on $C$, and pick any $Z \in |A|$. 
Then, by the commutative diagram on page \pageref{diagram}, we have that $L+K_S$ fails to be $k$-spanned on $Z$, whence $Z$ is contained in a divisor satisfying $(\dagger)$ for $k=k_0$. By our assumption, this yields $Z \sub D$ for some $D \in |-K_S|$. So we have $h^0((-K_S)_C-Z) > 0$, and $\deg ((-K_S)_C-Z)= 1$. So $(-K_S)_C-Z \sim y$, for some $y \in C$, and since
\[ h^0(Z) = h^0((-K_S)_C-y)= h^0((-K_S)_C)=2, \]
$y$ must be a base point of $|(-K_S)_C|$ and hence of $|-K_S|$. Hence $x=y$,
and $C$ passes through $x$.

This case will turn out to be the case (III) mentioned in the introduction.
We have left to prove that the numerical conditions  are equivalent to the fact that the only divisor satisfying $(\dagger)$ for $k=k_0$ is $-K_{S_8}$. This will be done in Section \ref{conc} below, since we first will need to study exceptional curves on Del Pezzo surfaces.

We close this section by giving the already mentioned example of Serrano of non-constant Clifford index and gonality.

\begin{example}[Serrano] \label{serrano}
 {\rm  Let $S$ be a Del Pezzo surface of degree $1$. Consider $L := -nK_S$ for an integer $n \geq 3$ and let $C$ be a smooth curve in $|L|$. If $C$ passes through the base point $x$ of $|-K_S|$, then $\O_C(-K_S-x)$ is a base point free $g^1_{n-1}$ on $C$, otherwise $\O_C(-K_S)$ is a base point free $g^1_n$ on $C$. Serrano shows that in both cases, these pencils compute the gonality and also the Clifford index (for more details see \cite[Ex.(2.2)]{par} and \cite[Ex.(3.15)]{ser}). }
\end{example}

We clearly see that in Serrano's example the only divisor class satisfying the conditions $(\dagger)$ for $k=n-2$ is $-K_S$.










\section{Exceptional curves} 
\label{exc}

We will in this section study the exceptional curves on a Del Pezzo surface. 
Clearly, on a Del Pezzo surface of type $S_n$, all the strict transforms of
the smooth plane curves are exceptional. The following is a non-trivial example of smooth curves of Clifford dimension $3$.

\begin{example} \label{excd3}
 {\rm  Let $S \iso S_6$ and consider the line bundle $L \sim -3K_{S_6}$. All the smooth curves in $|L|$ are of genus $10$, and for any such $C \in |L|$, we have $h^0(\O_C(-K_{S_6}))=h^1(\O_C(-K_{S_6}))=4$, so $\O_C(-K_{S_6})$ contributes to the Clifford index of $C$. We compute
\[ \Cliff \O_C(-K_{S_6}) = \deg \O_C(-K_{S_6}) -2(h^0(\O_C(-K_{S_6}))-1)=3. \]

Now assume $C$ is not exceptional of Clifford index $3$. Then $d := \gon C \leq 5$, and by 
Theorem \ref{birkvathm} (with the additional information of Proposition \ref{findD}) there has to exist an effective divisor $D$ satisfying $D.L-D^2=d$ and such that $D$ is of one of the following types:
\begin{itemize}
\item[(a)] $D^2=0$, $-D.K_{S_6}=2$,
\item[(b)] $D^2=1$, $-D.K_{S_6}=3$,
\item[(c)] $D \sim -K_{S_6}$,
\item[(d)] $D^2=-D.K_{S_6}=4$.
\end{itemize}
In these four cases we get respectively $D.L-D^2=-3D.K_{S_6}-D^2=6,8,6,8$, whence a contradiction.

Hence all the smooth curves in $|L|$ have gonality $6$ and Clifford index $3$, which shows that they are exceptional. If the Clifford dimension of any such $C$ is $2$, then $C$ is isomorphic to a smooth plane septic, and we easily get a contradiction from the genus formula of a smooth plane curve. So all the curves have Clifford dimension $3$.}
\end{example}

We show in this section that
the only exceptional curves on a Del Pezzo surface are precisely the strict transforms of the smooth plane curves and of the curves in the above example.

First we need the following result, which is similar to
\cite[(0.5.1)]{par}, but weaker and extended to Del Pezzo surfaces of degree one.

\begin{prop} \label{ggR}
  Let $S$ be a Del Pezzo surface and $R$ a vector bundle on $S$
  generated by its global sections apart from a finite scheme of
  length $n$. Then there exists a globally generated vector bundle $F$
  such that $\det F =\det R$ and 
  \begin{itemize}
  \item[(a)] $\rk F = \rk R+n$,
  \item[(b)] $c_2(F) = c_2(R)+m$, $n \leq m \leq 2n$,
  \item[(c)] $h^0(F \v) = h^0(R \v)$.
  \end{itemize}
\end{prop}

\begin{proof}
  We have an exact sequence
\[ 0 \hpil V \hpil H^0(R) \* \O_S \hpil R \hpil S_R \hpil 0, \]
 defining $V$ and $S_R$, and by hypothesis we have $\length S_R=n$.

  Dualising this sequence and tensoring by $\omega_S$, we get
\[ 0 \hpil R \v \* \omega_S \hpil H^0(R) \v  \* \omega_S 
        \hpil V \v  \* \omega_S \hpil \varepsilon_R \* \omega_S \hpil 0, \]
where $\varepsilon_R :=  \EExt ^2 (S_R, \O_S)$.

By Serre duality, the map 
$H^2 (R \v \* \omega_S) \khpil H^0(R) \v  \* H^2 (\omega_S)$ is an
isomorphism, whence the map $H^0 (V \v  \* \omega_S) \khpil H^0
(\varepsilon_R \* \omega_S)$ is surjective. Since $\omega_S \v$ has at
most one base point, it follows that the map $H^0 (V \v) \khpil H^0
(\varepsilon_R)$ is surjective off this base point. More precisely,
there is a torsion sheaf $\sigma$ only supported on this base point and
an exact sequence 
\[ H^0 (V \v) \* \O_S \hpil H^0(\varepsilon_R) \* \O_S \hpil \sigma
\hpil 0. \]
Clearly $\length \sigma \leq n$.

Now we proceed analogously to \cite[(1.5)]{gl} and
\cite[(0.5.1)]{par}.

Assume first $h^0(R \v)=0$. Then, since $V \v$ is generated by its
global sections, we can find a subspace $W$ of $H^0 (V \v)$ generating
$V \v$ and sitting in an exact sequence
\[ 0 \hpil H^0(R) \v \hpil W \hpil H^0(\varepsilon_R) \hpil  H^0(\sigma)
\hpil 0. \]

Now define $F$ to be the dual of the kernel of the evaluation map 
$W \* \O_S \khpil V \v$. By construction $F$ is generated by its global
sections.

Denoting by $Q$ the kernel of the map $V \v \khpil \varepsilon_R$, we
have a commutative diagram with exact rows and columns

\[ 
\xymatrix{
           & 0 \ar[d] & 0 \ar[d]  & & & 
\\
  0 \ar[r] & R \v  \ar[r] \ar[d] & F \v \ar^f[d] & & & 
\\
  0 \ar[r] & H^0(R) \v  \* \O_S \ar[r] \ar[d] & W \* \O_S \ar^g[r] \ar[d] & 
    H^0(\varepsilon_R) \* \O_S \ar[r]  & \sigma  \ar[r] & 0
\\ 
  0 \ar[r] & Q \ar[r] \ar[d] &  V \v \ar[r] \ar[d] & \varepsilon_R \ar[r] & 0 & 
  \\
  & 0 & 0.  & & &
} \]

Define $\alpha:= g \circ f$. By diagram-chase one easily sees that
$\sigma':= \coker \alpha$ sits in a short exact sequence
\[ 0 \hpil \varepsilon_R \hpil \sigma ' \hpil \sigma \hpil 0, \]
so $ n \leq \length \sigma ' \leq 2n$. The properties (a)-(c) are then
easy to check by the exact sequence
\[ 0 \hpil R \v  \hpil F \v \hpil H^0(\varepsilon_R) \* \O_S \hpil
\sigma' \hpil 0. \]

If $h^0 (R \v) >0$, we clearly have $R = R_0 \+H$ where $H$ is a
trivial vector bundle and $R_0$ is a vector bundle generated by its
global sections away from a finite set and such that $h^0 (R_0 \v)=0$.
Then we can apply the previous construction to $R_0$.
\end{proof}

We now make
the following assumptions:

\begin{itemize}
\item[(*)] Let $C_0$ be a smooth curve of genus $g \geq 4$ on a Del
  Pezzo surface. Set $L:= \O_S (C_0)$. Let $A$ be a base point free line 
bundle on $C_0$ such that $3 \leq h^0 (A) \leq h^1 (A)$ (this implies
$\deg A \leq g-1$) and $\omega_C \* A \v$ is base point free.
\end{itemize}

As in Subsection \ref{cliff} we get the vector bundle $E:=E(C_0,A)$,
which is of rank $h^0(A) \geq 3$.
Since by (\ref{eq:8}) $h^0(E \* \omega_S) >0$, we can find an
invertible subsheaf $0 \khpil M \khpil E$, and after saturating, we
can assume we have an exact sequence
\begin{equation}
  \label{eq:9}
0 \hpil M \hpil E \hpil R \hpil \tau \hpil 0,
\end{equation}
where $R$ is locally free of rank $\geq 2$ and $\tau$ is supported on a finite set.

Define $D:= \det R$. We have

\begin{lemma} \label{small}
  \begin{itemize}
  \item[(a)] $h^0 (R \v)=h^1 (R \v)=0$,
  \item[(b)] $D$ is non trivial and either $D \sim -K_{S_8}$ or $D$ is
    base point free. In particular $h^1(D)=h^2(D)=0$. Moreover $L \sim M+D$.
  \end{itemize}
\end{lemma}

\begin{proof}
  Similar to the proof of \cite[(1.12)]{par}.
\end{proof}

For any $C \in |L|$ we write $D_C := \O_C (D)$.

\begin{prop} \label{firstcliff}
        \begin{itemize}
  \item[(a)] $h^0 (D_C)$ is the same for any $C \in |L|$,
  \item[(b)] $D_{C_0} \geq A$,
  \item[(c)] $h^1 (D_C) = h^0 (M \* \omega_S) > 0$ for any $C \in |L|$.
  \item[(d)] $\Cliff D_C$ is the same for any $C \in |L|$ and $\Cliff D_C \leq \Cliff A$.
        \end{itemize}
\end{prop}

\begin{proof}
  The statements (a)-(c) follow as in the proof of \cite[(1.13)]{par}.
  To prove (d), assume $R$ is globally generated off a finite scheme of
  length $n$ and apply Proposition \ref{ggR} to find a globally
  generated vector bundle $F$ satisfying
\begin{eqnarray} 
    \label{eq:13} & \det F =D \\
    \label{eq:10} & \rk F = \rk R +n \geq 2, \\
    \label{eq:11} & c_2(F)=c_2(R)+m, \hs n \leq m \leq 2n, \\
    \label{eq:12} & h^0(F \v)=0.
\end{eqnarray}  

So $F$ is of the form $E(Q_0,B,V)$ for some $D_0 \in
|D|$, a line bundle $B$ on $D_0$ such that $B$ and $\N_{D_0/S} \* B
\v$ are base point free and $V$ is a subspace of $H^0(B)$ spanning
$H^0(B)$ and such that 
\begin{eqnarray}
  \label{eq:30a} \rk F = \dim V \leq h^0(B),\\
  \label{eq:30b} h^0(B)- \dim V=h^1(F \v),\\
  \label{eq:30c} c_2(F)=\deg B.
\end{eqnarray}

If $D^2 >0$, then $D_0$ is a smooth (irreducible) curve. If
$D^2=0$, then $D \sim r \Sigma$ for a smooth (irreducible) curve
$\Sigma$ and $r:= h^0(D)-1 \geq 1$, and $D_0$ consists of the disjoint
union of $r$ distinct smooth (irreducible) rational curves $\Sigma_1, \hdots,
\Sigma_r$ of self-intersection zero (see \cite[(1.4)]{gl}).

In any case let $\Cliff B = \deg B -2(h^0(B)-1)$.

We have

\begin{equation}
  \label{eq:16}
  \Cliff A \geq D.M + \Cliff B -2.
\end{equation}

Indeed, 
\begin{eqnarray*}
  \Cliff A = & c_2(E) -2(\rk E-1) & \mbox{by (\ref{eq:3}) and (\ref{eq:4})} \\
           = & D.M + c_2 (R) + \length \tau - 2\rk R & \mbox{by (\ref{eq:9})} \\
        \geq & D.M + c_2 (R) - 2\rk R & \\
           = & D.M + c_2 (F) - m - 2(\rk F-n) & \mbox{by
           (\ref{eq:10}) and (\ref{eq:11})} \\
        \geq & D.M + c_2 (F) - 2\rk F & \mbox{since $m \leq 2n$} \\
        \geq & D.M + \deg B - 2h^0(B) & 
          \mbox{by (\ref{eq:30a}) and (\ref{eq:30b})} \\
           = & D.M + \Cliff B -2.
\end{eqnarray*}

Grant for the time being the following

\vspace{.2cm}
{\bf Claim:}  (a) $\Cliff B=0$ or $1$, when $D \sim -K_{S_8}$,
  
(b) $\Cliff B=-2(r-1)$, for $r:=h^0(D)-1$, when $D^2=0$,

(c) $\Cliff B \geq 2g(D_0)-D^2$ otherwise.
\vspace{.2cm} 

If $D^2=0$, then by (\ref{eq:16}) and (b) of the claim:
\begin{equation}
  \label{eq:22}
  \Cliff A \geq D.L-2r.
\end{equation}
At the same time
\begin{equation}
  \label{eq:23}
h^0(D_C) = h^0(D)+h^1(-M) \geq r+1. 
\end{equation}
Combining (\ref{eq:22}) and (\ref{eq:23}) we get
the desired result $\Cliff A \geq \Cliff D_C$.   

Assume now $D^2 >0$. We have 
\begin{equation}
  \label{eq:17}
  h^0(D_C) = h^0(D)+h^1(-M) \geq h^0(D)= \frac{1}{2}D.(D-K_S)+1 = D^2
  -g(D_0) +2,
\end{equation}
whence
\begin{equation}
  \label{eq:18}
 \Cliff D_C = D.L - 2(h^0 (D_C)-1) \stackrel{(\ref{eq:17})}{\leq} D.M+2g(D_0)-D^2-2.
\end{equation}

If $D \sim -K_{S_8}$, then $h^0(D)=2$, and since $D_{C_0} \geq A$ and
$h^0(A) \geq 3$, we have $h^0(D_C)= h^0(D_{C_0}) \geq 3$, so we have strict inequalities in (\ref{eq:17}) and
(\ref{eq:18}).

Now the result follows by combining (\ref{eq:16}),
(\ref{eq:18}) and parts (a) and (c) of the claim.
\end{proof}
 
\renewcommand{\proofname}{Proof of claim}

\begin{proof}
If $D^2 =0$ and $D_0 = \Sigma_1 + \cdots + \Sigma_r$, then all the
$D_i$ are disjoint smooth rational curves and the line bundle $B$ on
$D_0$ is the direct sum $B= \+ B_i$ of line bundles $B_i$ on
$\Sigma_i$. Since both $B$ and $\N_{{D_0}/S} \* B \v$ are globally
generated, it follows that $B_i = \O_{\Sigma_i}$ for all $i$, so $\deg
B=0$ and $h^0(B)=r$ and $\Cliff B=-2(r-1)$ as asserted.

If $D \sim -K_{S_8}$, then $g(D_0)=1$ and since both $B$ and
$\N_{{D_0}/S} \* B \v$ are globally generated, we have $B = \O_{D_0}$ or $B= \N_{{D_0}/S}$,
so $\Cliff B=0$ or $\Cliff B=1$.

In all other cases we have $-K_S.D \geq 2$ by (P6), so $2g(D_0)-D^2=D.K_S+2
\leq 0$. If $B$ is special, then $\Cliff B \geq 0$ by Clifford's
theorem. If $B$ is non-special then by Riemann-Roch $\Cliff B
=2g(D_0)- \deg B$. Since $\N_{{D_0}/S} \* B \v$ is base point free, we
must have $\deg B \leq \deg \N_{{D_0}/S} =D^2$ and the result follows.
\end{proof}

\renewcommand{\proofname}{Proof}

We need the following easy 

\begin{lemma}  \label{cliffhanger}
  Let $L$ be a nef line bundle on a Del Pezzo surface with $g(L) \geq
  4$. Assume $L+K_S \sim D_1+ D_2$ for two divisors $D_1$ and $D_2$
  satisfying $h^0(D_i) \geq 2$. Then $\O_C(D_1)$ and $\O_C(D_2)$
  contribute to the Clifford index of any smooth $C \in |L|$ and
\[ \Cliff \O_C(D_1) = \Cliff \O_C(D_2) \leq D_1.D_2, \]
  with equality if and only if $h^0(\O_C(D_i))=h^0(D_i)$ and
  $h^1(D_i)=0$ for $i=1,2$. 

  Furthermore, if equality holds and there exists a smooth curve $C_0
  \in |L|$ such that $\Cliff C_0= D_1.D_2$, then any $(-1)$-curve
  $\Gamma$ with $\Gamma.L=0$ will satisfy $(\Gamma.D_1,\Gamma.D_2)=(0,-1)$ or $(-1,0)$.
\end{lemma}

\begin{proof}
  From the exact sequence
\[ 0 \hpil K_S -D_2 \hpil D_1  \hpil \O_C(D_1) \hpil 0, \]
$h^0(\O_C(D_1)) \geq h^0 (D_1) \geq 2$ and $h^1(\O_C(D_1)) \geq
h^2(K_S-D_2) \geq h^0 (D_2) \geq 2$, so that $\O_C(D_1)$ contributes
to the Clifford index of any smooth $C \in |L|$. By symmetry we get
the same result for $D_2$.

Now $\O_C(D_2) \iso \O_C(L+K_S-D_1) \iso \omega_C \* \O_C(D_1) \v$, so 
$\Cliff \O_C(D_1) = \Cliff \O_C(D_2)$. 

One computes by Riemann-Roch on $S$:
\begin{eqnarray*}
\Cliff \O_C(D_1) &  = & D_1.L-2(h^0(\O_C(D_1))-1) \leq D_1.L-2(h^0(D_1)-1) \\
               & \leq & D_1.L-D_1.(D_1-K_S) = D_1.(L-D_1+K_S) \\
                 &  = & D_1.D_2.
\end{eqnarray*}

It is clear that equality holds if and only if $h^0(\O_C(D_i))=h^0(D_i)$ and
  $h^1(D_i)=0$ for $i=1,2$.

For the last statement, since $-1= \Gamma.(L+K_S) = \Gamma.D_1 + \Gamma.D_2$, assume by symmetry to get a contradiction that $\Gamma.D_1
>0$. Since $-1= \Gamma.(L+K_S) = \Gamma.D_1 + \Gamma.D_2$, we get
$\Gamma.D_2=-1-\Gamma.D_1 \leq -2$. Clearly $h^0(D_1+ \Gamma) \geq
h^0(D_1) \geq 2$
and $h^0(D_2- \Gamma) = h^0(D_2) \geq 2$, so by what we have just
shown $\O_C(D_1+\Gamma)$ contributes
to the Clifford index of any smooth $C \in |L|$, and
\[ \Cliff \O_{C_0}(D_1+\Gamma) \leq  (D_1+\Gamma).(D_2-\Gamma)=
D_1.D_2 -2\Gamma.D_1 < \Cliff C_0, \]
a contradiction.
\end{proof}

We can now prove the following important result, whose proof is
unfortunately a bit long and tedious and involves the treatment of
several special cases.

\begin{prop} \label{long}
  Given the assumptions (*) and the exact sequence (\ref{eq:9}).
  Assume $C_0$ is exceptional and that $A$ computes the Clifford
  dimension of $C_0$.

  Then $h^0(R \* \omega_S)=0$.
\end{prop}

\begin{proof}
  Assume that $h^0(R \* \omega_S)>0$. Then we can put $R$ in a
  suitable exact sequence as in (\ref{eq:9}), namely
  \begin{equation}
    \label{eq:25}
0 \hpil P \hpil R \hpil R_1 \hpil \tau ' \hpil 0,     
  \end{equation}
where $P$ is a line bundle such that $P +K_S \geq 0$, $R_1$ is
locally free of rank $\rk R-1 \geq 1$ and $\tau '$ is of finite length.
Clearly $R_1$ is globally generated off a finite set. Set $Q:= \det R_1$.  Lemma \ref{small} applies for $Q$, which means that $h^1(Q)=h^2(Q)=0$, $Q >0$ and either $Q \sim -K_{S_8}$ or $Q$ is base point free.

Let $C \in |L|$. We get from
\begin{equation}
  \label{eq:25c}
0 \hpil -M-P   \hpil Q    \hpil Q_C   \hpil  0  
\end{equation}
that 
\begin{equation}
  \label{eq:26-}
  h^0(Q_C)=h^0(Q) + \delta,
\end{equation}
where we define
\begin{equation}
  \label{eq:d}
  \delta:= h^1 (-M-P)= h^1(L-Q+K_S).
\end{equation}

Clearly $h^0(Q_C) \geq 2$ and also $h^1(Q_C) = h^2(-M-P) = h^0
(L-Q+K_S) = h^0 (M+P+K_S) \geq 2$, since $M +P \geq -2K_S$, so 
$Q_C$ contributes to the Clifford index of any $C \in |L|$.

From (\ref{eq:9}) and (\ref{eq:25}) we get
\begin{eqnarray}
  \label{eq:25a} c_2(E)=D.M + c_2 (R)   + \length \tau, \\ 
  \label{eq:25b} c_2(R)=P.Q + c_2 (R_1) + \length \tau'
\end{eqnarray}

{\bf Case I:} $\rk R_1=1$.

We have $Q=R_1$ and $\rk E = h^0(A)=3$. Furthermore,
\begin{equation}
  \label{eq:26}
   \Cliff Q_C= \Cliff A +4+ \kappa,
\end{equation}
where
\begin{equation}
  \label{eq:27}
  \kappa:= -P.M + Q.K_S  - \length \tau - \length \tau' - 2\delta.
\end{equation}

Indeed, 
\begin{eqnarray*}
  \Cliff Q_C = & Q.L-2(h^0(Q_C)-1) &  \\
           = & Q.L-2(h^0(Q)-1) -2\delta   & \mbox{by (\ref{eq:26-})} \\
           = & Q.L-Q.(Q-K_S) -2\delta & \mbox{since $h^1(Q)=0$} \\
        = & Q.(M+P+K_S) -2\delta  & \\
        = & Q.(M+K_S) + c_2(E)-D.M - \length \tau - \length \tau'  -2\delta  & 
        \mbox{by (\ref{eq:25a}) and (\ref{eq:25b})}          \\       
 = & -P.M + Q.K_S+ \deg A - \length \tau - \length \tau'  -2\delta  & 
       \mbox{by (\ref{eq:3})} \\
   = & \Cliff A + 4 -P.M + Q.K_S- \length \tau - \length \tau' - 2\delta.
\end{eqnarray*}

Since $A$ computes the Clifford dimension of $C_0$, we must have
\begin{equation}
  \label{eq:27a}
 \kappa \geq -4 
\end{equation}


We now divide the proof in several subcases.

{\bf Case I-a:} $P.M=3$.

By (\ref{eq:26})-(\ref{eq:27a}) and (P6) we have $Q \sim -K_{S_8}$, $\delta=0$
and $\Cliff Q_{C_0}=\Cliff A$. Since $h^0(Q_{C_0})=h^0(Q)=2$, the
curve $C_0$ is not exceptional, a contradiction.

{\bf Case I-b:} $P.M=2$, $Q.K_S = -1$.

By (\ref{eq:26})-(\ref{eq:27a}) and (P6) we have $Q \sim -K_{S_8}$, $\delta=0$
and $\Cliff Q_{C_0} \leq \Cliff A +1$. As above, since $C_0$ is
exceptional, we must have equality in the last inequality, whence by
(\ref{eq:27}) we have $\length \tau = \length \tau'=0$. Since $Q$ has
one base point $x$, it follows that $R$ is not globally generated by
(\ref{eq:25}), whence by (\ref{eq:9}) and Lemma \ref{quick}, we have
that $C_0$ passes through $x$. 

Define the line bundle $A':= Q_{C_0}-x$ on $C_0$. Then clearly $A'$
contributes to the Clifford index of $C_0$, $h^0(A')=2$ and  
$\Cliff A' = \Cliff A$, again contradicting that $C_0$ is exceptional.

{\bf Case I-c:} $P.M=2$, $Q.K_S = -2$.
By (\ref{eq:26})-(\ref{eq:27a}) we have $\delta=\length \tau = \length
\tau'=0$, $h^0(Q_C)=h^0(Q)$ and $\Cliff A= \Cliff Q_{C_0}=Q.(M+P+K_S)$
(the last equality follows by Lemma \ref{cliffhanger}).

It is clear that $\O_{C_0}(P)$ contributes to the Clifford index of
$C_0$, and by Lemma \ref{cliffhanger}:
\begin{eqnarray}
  \nonumber  \Cliff \O_{C_0}(P) & \leq & P.(M+Q+K_S)=2+P.Q+P.K_S \\ 
  \label{eq:W}  & = & \Cliff A-Q.M-Q.K_S+P.K_S+2 \\
  \nonumber     & = & \Cliff A-Q.M+P.K_S+4.
  \end{eqnarray}

The same reasoning works for $M$, so we get
\begin{equation}
  \label{eq:A}
  Q.M \leq P.K_S +4 \hs \mbox{and} \hs Q.P \leq M.K_S +4.
\end{equation}

From (\ref{eq:25a}) and (\ref{eq:25b}), combined with $\deg A \leq
g(C_0)-1$, we get
\begin{equation}
  \label{eq:B}
  M.(M+K_S)+P.(P+K_S)+ Q^2 \geq 2.
\end{equation}
Furthermore, since $Q_C$ computes the Clifford index of $C_0$ and
$C_0$ is exceptional we must have $3 \leq h^0 (Q_C) = h^0(Q) =
\frac{1}{2}Q.(Q-K_S) +1 = \frac{1}{2}Q^2+2$, whence by the Hodge index
theorem on $Q$ and $-K_S$:
\begin{equation}
  \label{eq:C}
  Q^2 = \left\{ \begin{array}{ll}
             2 & \mbox{ if $S=S_8$ or $Q \sim -K_{S_7}$, } \\
             4 & \mbox{ if $Q \sim -2K_{S_8}$. }
          \end{array}
    \right . 
\end{equation}

If $P^2 \leq 0$, then $P.K_S \leq -2$ by (P6), and by (\ref{eq:A}) we have
$Q.M \leq 2$. If $Q.M =2$, then $\Cliff \O_{C_0}(P)  \leq \Cliff A$
by (\ref{eq:W}) and $h^0 (\O_{C_0}(P)) = h^0(P)=2$ by Lemma
\ref{cliffhanger}, contradicting the exceptionality of $C_0$ again.
So $Q.M=1$ and by the Hodge index theorem and (\ref{eq:C}) $M^2 \leq
0$. But then $M.K_S \leq -2$ and $M.(L-M+K_S) = M.(P+Q+K_S) \leq 2+1-2=1$, whence by 
Lemma \ref{cliffhanger} and the fact that $C_0$ is exceptional, we must have 
$M.K_S = -2$, $M^2=0$, $h^0(\O_{C_0}(M))=h^0(M)$ and $h^1(M)=0$. But then $h^0(\O_{C_0}(M))=2$ and $\O_{C_0}(M)$ computes the Clifford index $1$ of $C_0$, contradicting the exceptionality of $C_0$ again.

The same reasoning works if $P^2 \leq 0$.

This means that $P^2, M^2 \geq 1$.

If $P \sim -K_{S_8}$, then $\Cliff A = Q.(M+P+K_S)=Q.M$. At the same time, $M.(P+Q) = 2+M.Q$, and one can show that $M$ satisfies the conditions $(\dagger)$ for $k=1+M.Q$. Since $M$ clearly is not of any of the types (a)-(c) in Proposition \ref{constgonprop}, we have that all the smooth curves in $|L|$ have gonality $2+M.Q=\Cliff A +2$, whence $C_0$ is not exceptional, a contradiction.


% then $R$ is not globally generated by
% (\ref{eq:25}), whence by (\ref{eq:9}) and Lemma \ref{quick}, we have
% that $C_0$ passes through $x$, the base point of $|-K_{S_8}|$. 
% 
% As above define the line bundle $A':= P_{C_0}-x$ on $C_0$. Then clearly $A'$
% contributes to the Clifford index of $C_0$, and by (\ref{eq:W})
% $\Cliff A' = \Cliff P_{C_0}-1 \leq P.(M+Q+K_S)-1 =2$, whence all the
% smooth curves in $|L|$ are hyperelliptic, a contradiction.

The same reasoning works if $M \sim -K_{S_8}$.

So $P^2, M^2 \geq 1$ and $P.K_S, M.K_S \leq -2$. By (\ref{eq:A}) we also have $Q.M, Q.P \leq 2$.

We then easily see by the Hodge index theorem that the only solution to (\ref{eq:B}) and (\ref{eq:C}) is  $M \sim P \sim Q$ and $M^2=-M.K_S=2$. By (\ref{eq:W}) and Lemma \ref{cliffhanger} we get $\Cliff A = \Cliff \O_{C_0}(M)=M.(2M+K_S)=2$. At the same time $M.(L-M)=2M^2=4$, so $M$ satisfies the conditions $(\dagger)$ for $k=3$. Again $M$ is not of any of the types (a)-(c) in Proposition \ref{constgonprop}, so all the smooth curves in $|L|$ have gonality $4$, whence $C_0$ is not exceptional, a contradiction.

{\bf Case I-d:} $P.M \leq 1$.

By combining
Lemma \ref{cliffhanger} with (\ref{eq:26})-(\ref{eq:27a}) and (P6), and the fact that $h^1(Q)=0$, we get as above

\begin{equation} \label{eq:W'}
   \Cliff \O_{C_0}(P)  \leq \Cliff A +4-Q.M+P.K_S,
  \end{equation}
whence as above
\begin{equation}
  \label{eq:D}
  Q.M \leq P.K +4 \hs \mbox{and} \hs Q.P \leq M.K +4.
\end{equation}

As in case I-c we can show that $P^2, M^2 \geq 1$, which by the Hodge
index theorem gives that $M \sim P$ and $M^2=1$. By (\ref{eq:D}) we
have $Q.M \leq 3$. If equality occurs then $M \sim -K_{S_8}$ and
$\Cliff \O_{C_0}(M)= \Cliff A$ by (\ref{eq:W'}). By Lemma
\ref{cliffhanger} $h^0 (\O_{C_0}(M))=h^0(M)=2$, contradicting the
exceptionality of $C_0$. 

If $Q.M=2$, then by (\ref{eq:W'}) we have $-M.K_S \leq 2$, whence $M \sim P \sim -K_{S_8}$ and by (\ref{eq:W'}) again
\begin{equation}
  \label{eq:l1}
\Cliff \O_{C_0}(-K_{S_8})  \leq \Cliff A +1.  
\end{equation}
Since $-K_{S_8}.L=-K_{S_8}.(-2K_{S_8}+Q) =2+2=4$, $|-K_{S_8}|$ will induce a $g^1_4$ on every member of $|L|$, whence $\Cliff A=1$ and $C_0$ is a smooth plane quintic, so $\deg A=5$ and $g(L)=6$. 
Moreover, from (\ref{eq:26}) and (\ref{eq:27}) we see that $\delta =0$. 

We compute
\[ 10= L.(L+K_{S_8})=(-2K_{S_8}+Q).(-K_{S_8}+Q)= 8+Q^2, \]
whence $Q^2=2$. In particular, by Lemma \ref{small}, $Q$ is base point free, and so is $Q_{C_0}$.

Now $g(Q_0)=1$, whence $Q \geq -K_{S_8}$ and we have 
\[ Q \sim -K_{S_8}+ \Gamma, \]
for a $(-1)$-curve $\Gamma$. So 
\[ L \sim -3K_{S_8}+ \Gamma. \]

Since $C_0$ is a smooth plane quintic, it is well-known (see e.g. \cite[Beispiel 4]{ma1}) that we must have
\[ 2A \sim \omega_{C_0} \sim (L+K_{S_8})_{C_0} \sim (-2K_{S_8}+ \Gamma)_{C_0}, \]
whence $\O_{C_0} (\Gamma) \sim 2y$, for a point $y \in C_0$ (since $\deg \O_{C_0} (\Gamma)=\Gamma.L=2$), and we have
\[ A \sim \O_{C_0}(-K_{S_8})+y \sim Q_{C_0}-y. \]
Since $h^1(Q)=0$, $h^1(-2K_{S_8}+ \Gamma)=0$ and $\delta=0$, we get from (\ref{eq:26-}) and (\ref{eq:d}) that 
\[ h^0 (Q_{C_0})= \frac{1}{2}Q.(Q-K_{S_8})+1=3=h^0(A), \]
contradicting the base point freeness of $Q_{C_0}$.

If $Q.M \leq 1$, then either $P \sim M \sim Q \sim -K_{S_8}$ or $M.K_S = -3$. In the first case we get $L \sim -3K_{S_8}$, which belongs to the special case studied by Serrano and described in Example \ref{serrano}, where none of the smooth curves in $|L|$ are exceptional, a contradiction. In the second case, we get by Lemma \ref{cliffhanger}
\[ \Cliff \O_{C_0}(M) \leq M.(M+Q+K_S) \leq 1+1-3=-1, \]
a contradiction by Clifford's theorem.

{\bf Case II:} $\rk R_1 \geq 2$.

We have $h^0(A) \geq 4$.

By (\ref{eq:3}), (\ref{eq:4}), (\ref{eq:25a}) and (\ref{eq:25b})
\begin{eqnarray}
  \label{eq:pq} \Cliff A & = & c_2(E)-2(\rk E-1) = D.M + c_2(R) + \length
  \tau -2 \rk R \\ 
\label{eq:pqr} & = & D.M + P.Q + c_2 (R_1) -2 \rk R_1 -2 + 
    \length \tau + \length \tau'. 
\end{eqnarray}

Now $R_1$ is globally generated off a finite set of length $n \geq 0$,
and we can apply Proposition \ref{ggR} and find a globally generated
vector bundle $F$ such that $\det F =Q$ and
\begin{eqnarray}
  \label{eq:28} \rk F = \rk R_1+n, \\
  \label{eq:29} c_2(F)= c_2(R_1)+m, \hs n \leq m \leq 2n, \\
  \label{eq:30} h^0(F \v)= h^0(R_1 \v) = h^0(R \v) = h^0(E \v) = 0.
\end{eqnarray}

So $F$ is of the form $E(Q_0,B,V)$ for some $Q_0 \in
|Q|$, a line bundle $B$ on $Q_0$ such that $B$ and $\N_{Q_0/S} \* B
\v$ are base point free and $V$ is a subspace of $H^0(B)$ spanning
$H^0(B)$ and such that (\ref{eq:30a})-(\ref{eq:30c}) are satisfied.

As above, $Q_0$ is a smooth curve if $Q^2 >0$. If
$Q^2=0$, then $D \sim r \Sigma$ for a smooth (irreducible) curve
$\Sigma$ and $r:= h^0(Q)-1 \geq 1$, and $Q_0$ consists of the disjoint
union of $r$ distinct smooth (irreducible) rational curves $\Sigma_1, \hdots,
\Sigma_r$ of self-intersection zero.

We get
\begin{eqnarray*}
  \Cliff A = & D.M + P.Q + c_2 (F)-m  -2 (\rk F-n)  & \\  
             & -2 + \length \tau + \length \tau' &  \mbox{by (\ref{eq:pqr}),
    (\ref{eq:28}) and (\ref{eq:29})} \\
           \geq & D.M + P.Q + \deg B-m  -2h^0(B) +2n & \\  
             & -2 + \length \tau + \length \tau'   & 
                        \mbox{by (\ref{eq:30a}) and (\ref{eq:30c})} \\
    = & D.M + P.Q + \Cliff B -m +2n & \\  
             & -4 + \length \tau + \length \tau'  &  \\
    \geq  & M.P + M.Q + P.Q + \Cliff B -4  & \mbox{since $m \leq 2n$}.\\
\end{eqnarray*}

As in (\ref{eq:26}) we can prove by (\ref{eq:25c})-(\ref{eq:d}) that
\begin{equation}
  \label{eq:31-}
\Cliff Q_C = Q.(M+P) + 2g(Q_0)-2-Q^2-2\delta.  
\end{equation}

The claim in Proposition \ref{firstcliff} holds for $B$ and $Q$, and
arguing as in the sequel of that claim we get the two possibilities:
\begin{eqnarray}
  \label{eq:I}  Q \not \sim -K_{S_8} \hs \mbox{ and } \hs \Cliff A & \geq & P.M + \Cliff Q_C -2 + 2\delta, \hs
  \mbox{ or } \\
  \label{eq:II} Q \sim -K_{S_8} \hs \mbox{ and } \hs 
                \Cliff A & \geq & P.M + \Cliff Q_C -3 + 2\delta.
\end{eqnarray}

We see that we already get the desired contradictions if $P.M >2$ in 
(\ref{eq:I}) and if $P.M >3$ in (\ref{eq:II}). Also, if $P.M =3$ in
(\ref{eq:II}), we get $\Cliff A = \Cliff Q_C$ and $h^0(Q_{C_0}) =
h^0(Q) =h^0 (-K_{S_8})=2$, contradicting the exceptionality of $C_0$.

So we will from now on assume that $P.M \leq 2$.

Before we divide the rest of the proof in four special cases, we
deduce the following important information:
\begin{eqnarray}
  \label{eq:aa} h^0(M+P+K_S) = \frac{1}{2}M.(M+K_S) + \frac{1}{2}P.(P+K_S)
  + M.P + 1 + \delta \\
   \label{eq:bb} M.(M+K_S) \leq 0 \hs \mbox{ and } \hs P.(P+K_S) \leq 0.
\end{eqnarray}

Indeed, (\ref{eq:aa}) follows by Riemann-Roch. For (\ref{eq:bb}),
assume now, to get a contradiction, that $M.(M+K_S) \geq 2$. Then $h^0
(M+K_S) \geq 2$ and by Lemma \ref{cliffhanger}, $(P+Q)_{C_0}$
contributes to the Clifford index of $C_0$, and  
\begin{eqnarray*}
\Cliff (P+Q)_C & \leq & (P+Q).(M+K_S) = Q.(M+P+K_S)- Q.P + P.(M+K_S) \\
  & = & \Cliff Q_C + 2\delta + M.P -Q.P +P.K_S.
\end{eqnarray*}
By (\ref{eq:I}) and (\ref{eq:II}), we get 
\begin{equation}
  \label{eq:cc}
  \Cliff A \geq \left\{ \begin{array}{ll}
 \Cliff (P+Q)_C -2 + Q.P -P.K_S, & \mbox{ if $Q \not \sim -K_{S_8}$, } \\
 \Cliff (P+Q)_C -3 -2 P.K_{S_8},       & \mbox{ if $Q \sim -K_{S_8}$. }
          \end{array}
    \right . 
\end{equation}

If $Q \not \sim -K_{S_8}$,
(\ref{eq:cc}) gives that $Q.P -P.K_S \leq 2$, which means that either
$-P.K_S=2$ and $Q.P=0$, or $-P.K_S=1$ and $Q.P \leq 1$. In the first
case we get $P.(M+Q+K_S) \leq 2+0-2=0$, whence all the smooth curves
in $|L|$ would be hyperelliptic by Lemma \ref{cliffhanger}, a
contradiction on the exceptionality of $C_0$. In the second case, the
Hodge index theorem would give the contradiction $Q \sim P \sim -K_{S_8}$. 

If $Q \sim -K_{S_8}$, (\ref{eq:cc}) gives that $-P.K_{S_8} =1$, whence
$P \sim -K_{S_8}$ by (P6). Since $M.P = -M.K_{S_8}
\leq 2$, the Hodge index theorem together with the assumptions
$M.(M+K_S) \geq 2$, gives $M \sim -2K_{S_8}$. Hence $L \sim -4K_{S_8}$. But this case belongs to the special cases of Serrano
described in Example \ref{serrano} above, where we saw that none of the smooth
curves in $|L|$ were exceptional, a contradiction. 

This proves that $M.(M+K_S) \leq 0$, and by symmetry we obtain the
same result for $P$.

This shows (\ref{eq:bb}).

{\bf Case II-a:} $Q \not \sim -K_{S_8}$ and $P.M \leq 1$.

By (\ref{eq:I}), we have $\delta=0$ and by (\ref{eq:aa}) and
(\ref{eq:bb}), we must have, by the Hodge index theorem again, $M \sim
P \sim -K_{S_8}$, since clearly $h^0(M+P+K_S) \geq 2$. Now (\ref{eq:31-}) and
(\ref{eq:I}) again give
\begin{equation}
  \label{eq:40}
  \Cliff A \geq \Cliff Q_C-1 = -K_{S_8}.Q-1.
\end{equation}
But by Lemma \ref{cliffhanger} we have
\[ \Cliff (-K_{S_8})_{C_0} \leq -K_{S_8}.Q, \]
and since $C_0$ is exceptional we must have equality in (\ref{eq:40}),
whence also in (\ref{eq:I}). From (\ref{eq:9})  and (\ref{eq:25}) we
deduce
\begin{equation}
  \label{eq:46}
 \tau = \tau'= \emptyset, 
\end{equation}
and by part (c) in the claim in Proposition \ref{firstcliff}, we have
\begin{equation}
  \label{eq:47}
  \Cliff B = 2g(Q_0)-Q^2= Q.K_{S_8}+2 \hs \mbox{ if } \hs Q^2 >0.
\end{equation}
As argued above, $C_0$ cannot pass through the base point of
$|-K_{S_8}|$, so by Lemma \ref{quick}, $E$ is globally generated,
whence by (\ref{eq:46}), (\ref{eq:9})   and (\ref{eq:25}), we have
that $R$ and $R_1$ are also globally generated. By Proposition
\ref{ggR} we can assume that $F=R_1$. Therefore, by
(\ref{eq:8}), (\ref{eq:9}), (\ref{eq:25}) and (\ref{eq:46}), we get 
\begin{equation}
  \label{eq:48}
  h^1(B)=h^0(R_1 \* K_{S_8}) =h^0(E \* K_{S_8}) -2 = h^1(A)-2 \geq 2.
\end{equation}
So $B$ contributes to the Clifford index of $Q_0$ and if $Q^2 >0$ we
have by (\ref{eq:47}) and Clifford's theorem that $Q.K_{S_8} =-2$.
This last equality also holds if $Q^2=0$, whence by 
(\ref{eq:40}) we
therefore get $\Cliff A=1$, so $C_0$ is a smooth plane quintic with $h^0(A)=3$,
a contradiction.

{\bf Case II-b:} $Q \not \sim -K_{S_8}$ and $P.M =2$.

By (\ref{eq:I}), we have $\delta=0$ and $\Cliff A = \Cliff Q_{C_0}$, and by our assumptions that $A$ computes the Clifford dimension of
$C_0$ we have, by using (\ref{eq:aa})
\[ 4 \leq h^0(A) \leq h^1(Q_{C_0}) = h^0 (M+P+K_S) =
\frac{1}{2}M.(M+K_S) + \frac{1}{2}P.(P+K_S) +3, \]
whence
\begin{equation}
  \label{eq:42}
  M.(M+K_S) + P.(P+K_S) \geq 2,
\end{equation}
which is incompatible with (\ref{eq:bb}).

{\bf Case II-c:} $Q \sim -K_{S_8}$ and $P.M \leq 1$.

We first show that both $P^2 >0$ and $M^2 >0$. Assume to get a
contradiction that $P^2 \leq 0$. Then $P.(P+K_S) \leq -2$ and by
(\ref{eq:aa}) and (\ref{eq:bb}), combined with $h^0(M+P+K_S) \geq 2$,
we get $M.P + \delta \geq 2$,
and by (\ref{eq:II}) we get
\[ \Cliff A \geq \Cliff Q_{C_0} -P.M +1, \]
and this gives us the only possibility $P.M=\delta=1$.
But then $\Cliff A = \Cliff Q_{C_0}$ and by (\ref{eq:26-}) we get
$h^0(Q_C)=3 < h^0(A)$, contradicting the fact that $A$ computes the
Clifford dimension of $C_0$. The same reasoning works for $M$, so we
can conclude that $P^2 >0$ and $M^2 >0$.

By the Hodge index theorem we therefore must have $P \sim M$ and
$M^2=1$. If $M \not \sim -K_{S_8}$, we have $M.(M+K_S) \leq -2$ and by
(\ref{eq:aa}), again combined with $h^0(M+P+K_S) \geq 2$, we get
$\delta \geq 2$. But then (\ref{eq:II}) yields the contradiction
\[ \Cliff A \geq \Cliff Q_{C_0} +2. \]

Therefore we have $P \sim M \sim Q \sim -K_{S_8}$ and $L \sim -3K_{S_8}$, again a case belonging to Example \ref{serrano} above, a contradiction. 

{\bf Case II-d:} $Q \sim -K_{S_8}$ and $P.M =2$.

By (\ref{eq:II}) and the fact that $C_0$ is assumed to be exceptional,
we have $\delta=0$ and
\begin{equation}
  \label{eq:50}
\Cliff A = \Cliff Q_{C_0}-1 = -K_{S_8}.(L + 2K_{S_8})-1= -L.K_{S_8}-3, 
\end{equation}
where the middle equality follows from Lemma \ref{cliffhanger} and the
fact that $\delta=0$.

By assumption $h^1(A) \geq h^0(A) \geq 4$, so by (\ref{eq:007}) and 
(\ref{eq:50}), we get
\[ 8 \leq g(L)+1 -\Cliff A = \frac{1}{2}L.(L+K_{S_8}) +2 +L.K_{S_8}+3
= \frac{1}{2}L.(L+3K_{S_8}) + 5, \]
whence
\begin{equation}
  \label{eq:51}
 L.(L+3K_{S_8}) \geq 6. 
\end{equation}
 But then one easily computes
 \begin{eqnarray*}
 L.(L+3K_{S_8}) & = & (M+P-K_{S_8})^2 + 3K_{S_8}.(M+P-K_{S_8})\\
                & = & M.(M+ K_{S_8})+P.(P+K_{S_8})+2, 
 \end{eqnarray*}
and therefore (\ref{eq:51}) is incompatible with (\ref{eq:bb}).

This finally concludes the proof of Proposition \ref{long}.
\end{proof}

As an immediate consequence we get 

\begin{cor} \label{immediatecor}
Given the assumptions in Proposition \ref{long} and the exact sequence (\ref{eq:9}). We have
\[ D_{C_0} \sim A, \hs h^0(D_{C_0})=h^0(D) \hs \mbox{ and } \hs \Cliff C_0 =D.(L+K_S-D). \] 
\end{cor}

\begin{proof}
  Since $h^0(R \* \omega_S)=0$ by the last proposition, we get by Proposition \ref{firstcliff}(c) and 
  tensoring the sequence (\ref{eq:9}) with $\omega_S$ that
\[ h^1 (D_{C_0})=h^0(M+K_S)= h^0(E \* \omega_S)
  \stackrel{(\ref{eq:8})}{=} h^1(A) \geq 3, \]
whence $D_{C_0}$ contributes to the Clifford index of $C_0$ and we
must then have $\Cliff A = \Cliff D_{C_0}$ by Proposition
\ref{firstcliff}(d). Since we just saw that $ h^1 (D_{C_0})=h^1(A)$ it
follows from (\ref{eq:007}) that $ h^0 (D_{C_0})=h^0(A)$ also. Since
$D_{C_0} \geq A$ and $D_{C_0}$ is base point free (because it computes
the Clifford index of $C_0$), we must have $D_{C_0} \sim A$.

Finally, since $\Cliff A = \Cliff D_{C_0}$, one easily sees that we
must have equality in (\ref{eq:23}) and (\ref{eq:17}), whence $h^0(D_{C_0})=h^0(D)$ and
by Lemma \ref{cliffhanger} $\Cliff D_{C_0} = D.(L+K_S-D)$.
\end{proof}

Set $r:=h^0(A)-1$, which is by assumption the Clifford dimension of
$C_0$. We have then, by the last corollary, Riemann-Roch and the
adjunction formula: 
\begin{equation}
  \label{eq:52}
r= \frac{1}{2}D.(D-K_S) = D^2+1-g(D) = g(D)-1-D.K_S.  
\end{equation}

By \cite[Prop. 3.2]{elms} any pencil of divisors on $C_0$ has degree
$\geq 2r$. In particular, since $h^0 (\O_{C_0}(-K_S)) \geq h^0 (-K_S)
\geq 2$, we must have $-K_S.L \geq 2r = 2g(D)-2-2D.K_S$, which yields
\begin{equation}
  \label{eq:53}
  -M.K_S \geq -D.K_S + 2g(D)-2.
\end{equation}
Now if $g(D) \geq 2$, we have $h^0(D+K_S) \geq 2$, so by Lemma
\ref{cliffhanger}, $\O_{C_0}(D+K_S)$ contributes to the Clifford index
of $C_0$. Combining with (\ref{eq:53}), we get the contradiction
\[ \Cliff \O_{C_0}(D+K_S) \leq (D+K_S).M = D.M + M.K_S < D.M +D.K_S
= \Cliff C_0. \]
Hence
\begin{equation}
  \label{eq:54}
  g(D) \leq 1.
\end{equation}
By (\ref{eq:52}), this yields
\begin{equation}
  \label{eq:55}
r = \left\{ \begin{array}{ll}
             D^2+1 & \mbox{ if $g(D)=0$, } \\
             D^2 & \mbox{ if $g(D)=1$. }
          \end{array}
    \right .   
\end{equation}

Now define $d:=D.L$ and $c := \Cliff C_0 =D.(L+K_S-D)$. By \cite[Lemma
1.1]{elms}, the line bundle $A=D_{C_0}$ is very ample, whence it
embeds $C_0$ as a smooth curve $C_0'$ of degree $d$ and genus $g$ in $\PP^r$.
Moreover, since $H^0(D_{C_0}) \iso H^0(D)$, we have the following
commutative diagram, where we denote by $\phi_D$ and $\phi_{D_{C_0}}$
the morphisms defined by $D$ and $D_{C_0}$ respectively and set
$S':=\phi_D (S)$:
\begin{equation}
  \label{eq:commu}
\xymatrix{
    S  \ar[r]^{\phi_D}  &  S'  \ar[r]  & \PP ^r  \\ 
    C_0  \ar[r]^{\phi_{D_{C_0}}} \ar[u]  &  C_0' \ar[u]   \ar[r] &
    \ar@{=}[u] \PP ^r.  
} 
\end{equation}
We therefore have $\deg S' \leq D^2$. By \cite[Cor. 3.5 and Prop.
5.1]{elms}, a curve of Clifford dimension $r \geq 4$ in $\PP^r$ of
degree $d$ and genus $g$ with $(d,g) \not = (4r-3,4r-2)$ cannot be
contained in a (possibly singular) surface of degree $\leq 2r-3$. By (\ref{eq:55}) we get
\[ \deg S' \leq D^2 \leq r \leq 2r-3 \]
(when $r \geq 4$). So we therefore must have
\begin{equation}
  \label{eq:56}
 (d,g) = (4r-3,4r-2) \hs \mbox{ if } \hs r \geq 4.
\end{equation}

Now we make use of the following result:

\begin{prop} \label{conjelms}   {\rm \cite{elms}}
  Let $C$ be a smooth curve of genus $g$, Clifford index $c$, Clifford dimension $r \geq 3$ and
  degree $d$ in $\PP^r$. Consider the following properties:
  \begin{itemize}
  \item[(a)] $(d,g) = (4r-3,4r-2)$ (whence $c=2r-3$),
  \item[(b)] $C$ has a unique line bundle $A$ computing $c$,
  \item[(c)] $A ^2 \iso \omega_C$ and $A$ embeds $C$ as an
    arithmetically Cohen-Macaulay curve in $\PP^r$,
  \item[(d)] $C$ is $2r$-gonal, and there is a one-dimensional family
    of pencils of degree $2r$, all of the form $|A-B|$, where $B$ is
    the divisor of $2r-3$ points of $C$.
  \end{itemize}

If $r \leq 9$, then $C$ satisfies (a)-(d).

In general, if $C$ satisfies (a), it also satisfies (b)-(d).
\end{prop}

\begin{rem}
  {\rm In \cite{elms} it is furthermore conjectured that any curve of
  Clifford dimension $r \geq 3$ satisfies (a)-(d). }
\end{rem}

By this result and (\ref{eq:56}) we have
\begin{lemma} \label{wrong}
  $C_0$ satisfies the conditions (a)-(d) in Proposition \ref{conjelms}
  if $r \geq 3$.
\end{lemma}

By condition (c) we have $\O_{C_0}(2D) \iso \omega_{C_0} \iso
\O_{C_0}(L+K_S)$, so we have an exact sequence
\[ 0 \hpil K_S-2D \hpil L+K_S-2D \hpil \O_{C_0}  \hpil 0, \]
from which we get $h^0(L+K_S-2D)=1$. This means that
\begin{equation}
  \label{eq:57}
L+K_S-2D =: \Delta \geq 0,  
\end{equation}
where $\Delta.L=0$ and $\Delta$ is only supported on $(-1)$-curves.

Furthermore, from Lemma \ref{cliffhanger} and the fact that $D$ is nef (Lemma \ref{small}(b)), we must have $\Delta.D=0$, whence
by (\ref{eq:57}) $\Delta^2=\Delta.K_S$. Riemann-Roch therefore yields
\begin{equation*}
  h^1(\Delta)= h^0(\Delta)- \frac{1}{2}\Delta.(\Delta-K_S)-1=0.
\end{equation*}
From the exact sequence
\[ 0 \hpil K_S-\Delta \hpil 2D \hpil \O_{C_0}(2D)  \hpil 0, \]
we therefore get
\begin{equation}
  \label{eq:58}
  h^0(2A) =h^0(\O_{C_0}(2D))= h^0(2D)= D.(2D-K_S)+1=2r+D^2 +1.
\end{equation}
By \cite[Cor. 3.4]{elms}, we must have $h^0(2A) \geq 4r-2$
if $r \geq 3$. Combining (\ref{eq:55}) and (\ref{eq:58}) yields
\[ 4r-2 \leq h^0(\O_{C_0}(2D)) =2r+D^2+1 \leq 3r +g(D), \]
whence $r=D^2=3$ and $g(D)=1$.

We have therefore proved that an exceptional curve on a Del Pezzo surface has Clifford dimension $2$ or $3$.

\subsection{Curves of Clifford dimension $2$}

From  (\ref{eq:55}), we 
see that $D^2=2$ if $g(D)=1$. Hence $c=D.L-D^2+D.K_S=d-4$. On the
other hand, by Proposition \ref{fail}, we have that the
minimal gonality of a smooth curve in $|L|$ is $\leq D.(L-D)=d-2$.
Since $D$ is not of any of the types (a)-(c) in Proposition \ref{constgonprop} , all the smooth curves in $|L|$ have the same gonality, whence $C_0$ has gonality $d-2$, contradicting
its exceptionality. So we must have $(r,g(D),D^2)=(2,0,1)$.

By Proposition \ref{easyapp} we have that $\phi_D$ is an isomorphism
    ouside of finitely many contracted $(-1)$-curves. In particular it
    is birational.

By the commutative diagram (\ref{eq:commu}) with $r=2$, we have that
$\phi_D:S \khpil \PP^2$ is a blow up at finitely many points, which is
equal to the number $n(D)$. Clearly $0 \leq n(D) \leq 8$ and $S \iso
S_{n(D)}$.

Let
\begin{equation}
  \label{eq:60}
\R(D)= \{ \Gamma_1, \hdots, \Gamma_ {n(D)} \}. 
\end{equation}
Since $D_{C_0}$ is very ample, we must have
\begin{equation}
  \label{eq:61}
  C_0 \sim dD - \sum_{i=1}^{n(D)}a_i \Gamma_i,
\end{equation}
with all $a_i=0$ or $1$. In other words $\Gamma_i.L=0$ or $1$ for all
$i$. This means that $D_C$ is very ample for any smooth $C \in |L|$, so
all such are isomorphic to a smooth plane curve of degree $d$ and are
exceptional.

\subsection{Curves of Clifford dimension $3$}

From  (\ref{eq:55}), we have $D^2=3$ and $g(D)=1$, whence $-D.K_S=3$. By the Hodge index theorem $K_S^2 \leq 3$ with equality if and only if $D \sim -K_{S_6}$. It is known from the results of G.~Martens in \cite{ma} (and also contained in Proposition \ref{conjelms}(a)) that a curve $C_0$ of Clifford dimension $3$ satisfies
\begin{equation}
  \label{eq:n1}
  g(C_0)=10, \hs c:= \Cliff C_0=3, \hs d:=D.L=9 \hs \mbox{and} \hs (2D)_{C_0} \sim \omega_{C_0}.
\end{equation}

The Hodge index theorem gives $3L^2 = D^2L^2 \leq (D.L)^2=81$, whence $L^2 \leq 27$ with equality if and only if $L \sim 3D$. Since $g(L)=10$, we have $L.(L+K_S)=18$, and combined with the Hodge index theorem $K_S^2L^2 \leq (L.K_S)^2=(L^2-18)^2$, we end up with the following possibilities:
\begin{itemize}
\item[(i)]   $L^2=23$, $-L.K_S=5$, $K_S^2=1$.
\item[(ii)]  $L^2=24$, $-L.K_S=6$, $K_S^2=1$.
\item[(iii)] $L^2=25$, $-L.K_S=7$, $K_S^2=1$.
\item[(iv)]  $L^2=26$, $-L.K_S=8$, $K_S^2=1,2$. 
\item[(v)]   $L^2=27$, $-L.K_S=9$, $K_S^2=1,2,3$ ($K_S^2=3$ implying $L \sim 3D \sim -3K_{S_6}$).
\end{itemize}

We now show that we can rule out the two first cases.

Indeed, in case (i), the pencil $|-K_{S_8}|$ will cut out a $g^1_5$ on each member of $|L|$, in particular on $C_0$, contradicting its exceptionality.

In case (ii) we calculate
\[ \chi (L-2D) = \frac{1}{2}(L^2-4L.D+4D^2-K_S.L+2D.K_S)+1 = 1, \]
whence either $L-2D \geq 0$ or $K_S-L+2D \geq 0$. But $(K_S-L+2D).L =-12$, and since $L$ is nef, we must have $L-2D \geq 0$. Since $(L-2D).K_S=0$, we must have $L \sim 2D$, but this contradicts $(L-2D).L=6$.

We now consider the cases (iii)-(v).

In case (iii) we calculate $(L+K_{S_8}-2D)^2=0$ and $(L+K-2D).K_{S_8}=0$, whence by the Hodge index theorem $L \sim 2D-K_{S_8}$. We must have by Remark \ref{redundant} that $D \sim -K_{S_8} + \Gamma_1 + \Gamma_2$, with $\Gamma_1, \Gamma_2 \in \R_1(L)$, $\Gamma_1.\Gamma_2=0$ and  $\R(L) = \emptyset$. This means that we have
\begin{equation}
  \label{eq:n4}
  L \sim -3K_{S_8} + 2\Gamma_1 +2\Gamma_2, \hs \Gamma_1, \Gamma_2 \in \R_1(L),
\hs \Gamma_1.\Gamma_2=0, \hs \R(L) = \emptyset.
\end{equation}

In case (iv) we calculate $(L+K_S-2D)^2=K_S^2-2$ and $(L+K_S-2D)(-K_S)=2-K_S^2$.

If $K_S^2=1$, we calculate
$\chi (L+K_{S_8}-2D)=1$, whence either $L+K_{S_8}-2D \geq 0$ or $2D-L \geq 0$. But the latter is impossible, since $(-K_{S_8})(2D-L)=-1$. So 
$L+K_{S_8}-2D \geq 0$, and since $(L+K_{S_8}-2D)(-K_{S_8})=1$, we have $L+K_{S_8}-2D \sim \Gamma$ for a $\Gamma \in \R(L)$, by (\ref{eq:n1}). By  Remark \ref{redundant} we must have $D \sim -K_{S_8} + \Gamma_1 + \Gamma_2$, with $\Gamma_1.\Gamma_2=0$, $\R(L) = \{ \Gamma_1 \}$ and $\Gamma_2 \in \R_1(L)$. Therefore $\Gamma_1=\Gamma$ and we have
\begin{equation}
  \label{eq:n5}
 L \sim -3K_{S_8} + 3\Gamma_1 +2\Gamma_2, \hs \R(L) = \{ \Gamma_1 \}, \hs \Gamma_2 \in \R_1(L),
\hs \Gamma_1.\Gamma_2=0.
\end{equation}

If $K_S^2=2$, then we get as in case (iii) that $L \sim 2D-K_S$. By  Remark \ref{redundant} again, $D \sim -K_{S_7} + \Gamma$, with $\Gamma \in \R_1(L)$ and  $\R(L) = \emptyset$. Therefore
\begin{equation}
  \label{eq:n6}
 L \sim -3K_{S_7} + 2\Gamma, \hs  \Gamma \in \R_1(L), \hs \R(L) = \emptyset.
\end{equation}

In the same way we can treat the case (v) and we leave it to the reader to verify that we end up with the following cases:
\begin{eqnarray}
  \label{eq:n7} L & \sim  & -3K_{S_8} + 3\Gamma_1 +3\Gamma_2, \hs \R(L) = \{ \Gamma_1, \Gamma_2 \},
\hs \Gamma_1.\Gamma_2=0, \\
  \label{eq:n8} L  & \sim  & -3K_{S_7} + 3\Gamma, \hs \R(L) = \{ \Gamma \}, \\
  \label{eq:n9} L  & \sim  & -3K_{S_6}.
\end{eqnarray}

We now claim that any smooth $C \in |L|$ for an $L$ as in (\ref{eq:n4})-(\ref{eq:n9}) is exceptional. Note that all the smooth curves in $|L|$ for $L$ as in 
(\ref{eq:n4})-(\ref{eq:n8}) are strict transforms of some smooth curve in $|-3K_{S_6}|$, for a blowing up of $S_6$ in one or two points (in general position), so what we claim now follows from Example \ref{excd3} above.

We have therefore proved:

\begin{thm} \label{exccurves}
  Let $C_0$ be a smooth curve of Clifford index $c$ and
  Clifford dimension $r \geq 1$ on a Del Pezzo surface $S$. Let $A$ be
  a line bundle computing its Clifford dimension. Set $L := \O_S(C_0)$.

  Then $C_0$ is exceptional if and only if it is as in one of the following two cases:
  
  \begin{itemize}
  \item[(a)] There exists a base point free line bundle $D$ on $S$
  satisfying  
  \begin{eqnarray}
    \label{eq:properties} & D^2=1, \hs D.K_S=-3, \hs D.L=c+4 \hs 
              \mbox{ and } \\
\nonumber  &  \Gamma_i.C \leq 1 \hs \mbox{ for all } \Gamma_i \in \R(D)
  \end{eqnarray}
 (the latter condition is equivalent to $D_C$ being very ample for all $C \in |L|$). 

In this case $r=2$, $A \sim \O_{C_0}(D)$ and $\phi_D:S \khpil \PP^2$ gives the identification $S \iso S_{m(L)}$ and maps every smooth curve in $\O_S(C)$ to a smooth plane
  curve of degree $D.L=c+4$, whence all these curves are exceptional
  of Clifford dimension $2$ and Clifford index $c$.
  \item[(b)] $S \iso S_n$ for $n \in \{6,7,8 \}$ and 
\[ L \sim -3K_{S_n} + \sum _{i=1}^{n-6} a_i \Gamma_i, \hs \Gamma_i=(-1)\mbox{-curve}, \hs a_i \in \{2,3 \}. \]

In this case $r=3$, $g(L)=10$, $c=3$ and $A \sim \O_{C_0}(D)$, for 
\begin{equation}
  \label{eq:ddd}
  D:= -K_{S_n} + \sum _{i=1}^{n-6} \Gamma_i
\end{equation}
\end{itemize}
and all the smooth curves in $|L|$ are exceptional of Clifford dimension $3$ and Clifford index $3$.
\end{thm}

This concludes the proof of part (c) of Theorem \ref{gonconst}.



It is clear that after a change of basis of $\Pic S$ we can assume that that we have
\[ L \sim dl-\sum _{i=1}^{n} a_i e_i, \hs a_i \in \{1,2 \}, \hs \mbox{and} \hs D \sim l, \]
in case (a), and that we have 
\[ L \sim 9l-\sum _{i=1}^{6} e_i -\sum _{i=7}^{n} (3-a_i) e_i, \hs 
a_i \in \{2,3 \}, \hs \mbox{and} \hs D \sim 3l- \sum _{i=1}^{6} e_i \]
in case (b).



  \begin{rem}
    {\rm It is classically known that an exceptional curve of Clifford
    dimension $2$ and Clifford index $c$ has a one dimensional family
    of pencils computing the gonality $c+3$, all obtained by
    projecting from points on the curve. One can see this in our
    concrete example, since for any $x \in C$, we have $h^0(D_C-x)=2$
    and $\deg (D_C-x)=D.L-1=c+3$. It also means that any pencil
    computing the gonality of $C$ is of the form $D_C-x$ for $D$ a
    divisor satisfying (\ref{eq:properties}) and $x$ a point of $C$.

    By \cite{ma} (see also Proposition \ref{conjelms}(d) above), a similar result is    also known for an exceptional curve of Clifford
    dimension $3$, which in our case reads that any pencil
    computing the gonality of $C$ is of the form $D_C (-Z)$, where $D$ satisfies (\ref{eq:ddd}) and $Z$ is a length $3$ scheme on $C$ where $D$ fails to be $2$-very ample (cf. the proof of Proposition \ref{constgon}).}
  \end{rem}



\section{Conclusion of the proof of Theorem \ref{gonconst}}
\label{conc}


We have left to prove that the case (c) in Proposition \ref{constgonprop} is
Case (III) in the introduction. We have to prove the numerical
conditions, which follow by:

\begin{lemma}
  Let $L$ be a base point free line bundle on a Del Pezzo surface of
  degree $1$ such that $g(L) \geq 2k_0+1$ for an integer $k_0 \geq 1$. 
The following conditions are equivalent:
\begin{itemize}
\item[(i)] the only divisor satisfying $(\dagger)$ for $k=k_0$ 
           is $-K_S$,
\item[(ii)]$-L.K_S=k_0+2$, $L$ is ample, $L.C \geq k_0+2$ for all smooth 
           rational curves $C$ with $C^2 =0$,  $L.C \geq k_0+3$ for all smooth 
           rational curves $C$ with $C^2 =1$ if $(k_0,L^2)=(2,8)$, and $L.\Gamma \geq 2$ for all $(-1)$-curves 
           $\Gamma$ if $L^2 \geq 8$.
\end{itemize}
\end{lemma}

\begin{proof}
A divisor $D$ satisfying $(\dagger)$ for $k=k_0$ must be of one of the types
(a) or (b) with $g(D)=0$ listed in Proposition \ref{findD}, or of the
form (\ref{eq:form}) with $g(D)=1$. 

If there exists a $(-1)$-curve $\Gamma$ such that  
$\Gamma.L=1$, define $D:=-K_S+\Gamma$ and 
  $M:=L-D = L+ K_S - \Gamma$. If $L^2 \geq 8$, one easily computes
  $h^0(L-2D) \geq \frac{1}{2}(L^2-3k-6)+1 >0$ and $L.D-D^2=k_0+1$, so
  $D$ will satisfy the conditions $(\dagger)$ for $k=k_0$. If $L^2=7$
  (so that $k_0=1$), we find $(-2K_S+\Gamma) ^2= (-2K_S+\Gamma).L=7$,
  whence by the Hodge index theorem $L \sim -2K_S+\Gamma$, and the
  only divisor satisfying $(\dagger)$ for $k=k_0$ is $-K_S$.

So (i) is clearly equivalent to 
\begin{eqnarray}
  \label{eq:rep}
  & -L.K_S=k_0+2, \hs L \mbox{ is ample,} \hs L.\Gamma \geq 2 
\mbox{ for all $(-1)$-curves $\Gamma$ if } L^2 \geq 8, \\
\nonumber & L.D \geq D^2+k_0+2 \mbox{ for all smooth rational curves $D$ with $D^2
  =0$ or $1$. } 
\end{eqnarray}

We now show that the condition that $L.D \geq k_0+3$ for all smooth
rational curves $D$ with $D^2=1$, is redundant, except when $L^2=8$.

Indeed, assume that there is such a curve $D$ and that the conditions (ii)
are given. Then the only divisors satisfying  $(\dagger)$ for $k=k_0$ 
are  $-K_S$ and $D$. If $(k_0, L^2) \not = (2,8)$, then $(k_0, g(L)) \not =(2,3)$,
whence by Proposition \ref{findD} all the smooth curves in $|L|$ are
exceptional, and 
$\Gamma_i.L \leq 1$ for any $\Gamma_i \in \R(D)$. (Note that $\R(D) \not = \emptyset$, since otherwise $D$ is very ample and $S \iso \PP^2$ by 
Theorem \ref{exccurves}(a).) This clearly contradicts
the conditions (ii), unless possibly if $L^2=7$ and $\Gamma_i.L =1$
for all $i$. But we just saw above that in this case $L \sim -2K_S+\Gamma$ for a
$(-1)$-curve $\Gamma$, so clearly $\Gamma_i.L =2$ whenever $\Gamma_i
\not = \Gamma$.

Therefore, the conditions (i) and (ii) above are equivalent. 
\end{proof}

This concludes the proof of parts (a) and (b) of Theorem \ref{gonconst}.


\section{Pencils computing the gonality}
\label{pencilssection}

In Theorem \ref{exccurves} we described the line bundles $A$ computing the Clifford dimension of smooth exceptional curves on a del Pezzo surface. In particular, we saw that such a line bundle is the restriction of a line bundle on the surface.

We conclude this paper by describing the pencils computing the gonality of a smooth curve on a Del Pezzo surface. This will make some of the results of Pareschi more precise (see \cite [Lemma (2.8)]{par}), and extend them to Del Pezzo surfaces of degree $1$.

By Proposition \ref{firstgon}, if $A$ computes the gonality of a smooth curve $C$ on a Del Pezzo surface $S$, then there exists a line bundle $D$ on $S$ such that $\O_C(D) \geq A$. Furthermore, if the gonality is the minimal among the smooth curves in $|C|$ (which is always the case, except possibly for the cases (I), (II) and (III) described above), the line bundle $D$ has to be as in (a)-(f) of Proposition \ref{findD}. Furthermore, by Proposition \ref{findD} and Theorem \ref{exccurves} combined, if $D$ is as in (e), then $(\gon C,g (C))=(4,5)$, $(3,4)$ or $(6,10)$. 

In the special cases (I), (II) and (III), if $C$ has minimal gonality $d-1$, then we have seen that $D \sim -K_{S_7}$, $D \sim -K_{S_8} + \Gamma$ (where $\Gamma$ is a $(-1)$-curve) and $D \sim -K_{S_8}$ respectively, and there are no other line bundles satisfying $(\dagger)$ for $k=d-2$ by Proposition \ref{constgonprop}. If $C$ has gonality $d$, a careful analysis is needed. Denoting by $A$ again a line bundle computing the gonality, by Proposition \ref{firstgon} there exists an effective divisor $D'$ on $S$ satisfying $(\dagger)$ for $k=d-1$ (in particular $D'.L -{D'}^2 \leq d$) such that $\O_C(D') \geq A$. If $D' \not \sim D$, then we must have $D'.L -{D'}^2 = d$. We leave it to the reader to verify that $D'$ must also be nef.

We now treat the three different special cases and will find out that such a divisor $D'$ does not exist in the cases (I) and (II), and that there are a few possibilities in case (III).

{\bf Case (I):} We have $d=3$ and  $L \sim -2K_{S_7}$. In particular $-2K_{S_7}.D'=D'.L={D'}^2+3$. Since $L-D' \geq -K_S$, we have 
$D.L \leq (L+K_S).L=4$, whence $L.D'=4$, $D^2 =1$ and $-K_{S_7}.D'=2$, a contradiction on the parity.

{\bf Case (II):}
This case is similar to case (I) and left to the reader.

{\bf Case (III):}
We have $\deg S=1$ and $-L.K_{S_8}=d \geq 3$. By (\ref{eq:65b}) one sees that if $g(D') \geq 2$, then $D' \sim -2K_{S_8}$. One has $d=D'L-{D'}^2=-2K_{S_8}.L-4K_{S_8}^2 =2d-4$, whence $d=4$. We have $h^0(D'_C)=4$ and $\deg D'_C =8$, so there exists an effective divisor $Z$ of degree $4$ on $C$ such that $A \sim D'_C-Z$. This implies that $h^0(D' \* \I_Z)=h^0(A)=2$, so there are two distinct elements $D_1$ and $D_2$ in $|D|$ passing through $Z$. Since the only reducible elements of $|D'|$ are of the form $H_1+H_2$ for $H_i \in |-K_{S_8}|$ and $C$ does not pass through the base point of $|-K_{S_8}|$, one easily sees that $D_1$ and $D_2$ must be reduced and irreducible, in particular they intersect properly. Since $D_1.D_2=\deg Z=4$, we have $Z = D_1 \cap D_2$.

Conversely, picking any two irreducible elements $D_1$ and $D_2$ in $|-2K_{S_8}|$ intersecting in $4$ distinct points and setting $Z := D_1 \cap D_2$, one sees from the exact sequence
\[ 0 \hpil C-D_1 \hpil C \* \I_Z \hpil (C-D_1)_{D_2} \hpil 0 \]
that $|C \* \I_Z|$ is nonempty and has no base points off $Z$, whence from Bertini's theorem the generic member is smooth. For any such smooth $C'$, the line bundle $ (-2K_{S_8})_{C'}-Z$ is then a $g^1_d$.

Assume from now on that $g(D')=0$ or $1$. By computing $\Cliff D'_C$ as in the proof of Theorem \ref{findD}, we leave it to the reader to check that we end up with only three possibilities:
\begin{itemize}
\item[(1)] $g(D')=0$, ${D'}^2=0$,
\item[(2)] $g(D')=0$, ${D'}^2=1$, $d=3$,
\item[(3)] $g(D')=1$, ${D'}^2=2$.
\end{itemize}

Summarizing, we obtain the following complete classification:

\begin{prop}  \label{allpencils}
  Let $C$ be a smooth curve of genus $g \geq 2$ on a Del Pezzo surface and $A$ a line bundle computing its gonality.

(a)  If $C$ is of gonality $d$ and not as in the cases (I),(II) and (III), then $A$ is of one of the following types:
\begin{itemize}
\item[(i)]    $A=D_C$, for $D^2=0$, $-D.K_S=2$,
\item[(ii)]   $A=D_C-P$, $P$ any point of $C$, $D^2=1$, $-D.K_S=3$,
\item[(iii)]  $A=(-K_{S_8})_C-x$, $x$ base point of $|-K_{S_8}|$,
\item[(iv)]   $A=D_C-Z_2$, $D^2=-D.K_S=2$, $Z_2$ a length $2$ scheme where $D$ fails to be very ample, $K_S^2 \leq 2$,
\item[(v)]    $A=D_C-Z_3$, $D^2=-D.K_S=3$, $Z_3$ a length $3$ scheme where $D$ fails to be $2$-very ample, $(d,g)=(3,4)$, $(4,5)$ or $(6,10)$, $K_S^2 \leq 3$,
\item[(vi)]  $A=D_C-Z_4$, $D^2=-D.K_S=4$, $Z_4$ a length $4$ scheme where $D$ fails to be $3$-very ample, $(d,g)=(4,5)$, $K_S^2 \leq 4$.
\end{itemize}

(b) If $C$ is as in case (I) (resp. (II)), then $A = (-K_{S_7})_C-Z_2$ (resp. $A = (-K_{S_7}+ \Gamma)_C-Z_2$, with $\Gamma$ a $(-1)$-curve), with $Z_2$ a length $2$ scheme where $-K_{S_7}$ (resp. $-K_{S_7}+ \Gamma$) fails to be very ample if $C$ is hyperelliptic; and $A = (-K_{S_7})_C-P$ (resp. $A = (-K_{S_7}+ \Gamma)_C-P$) for any point $P \in C$ if $C$ is trigonal.

(c) If $C$ is as in case (III) and of minimal gonality $d-1$, then $A = (-K_{S_8})_C-x$, where $x$ is the base point of $|-K_{S_8}|$. If $C$ has gonality $d$ (equivalently $C$ does not pass through $x$), then $A$ is of one of the following types:
\begin{itemize}
\item[(i)]     $A = (-K_{S_8})_C$,
\item[(ii)]    $A = (-2K_{S_8})_C-Z_4$, where $Z_4$ has length $4$ and $Z_4 = D_1 \cup D_2$ for two distinct curves $D_1$ and $D_2$ in $|-2K_{S_8}|$,
\item[(iii)]    $A=D_C$, for $D^2=0$, $-D.K_S=2$,
\item[(iv)] $A=D_C-P$, $P$ any point of $C$, $D^2=1$, $-D.K_S=3$, $d=3$,
\item[(v)] $A=D_C-Z_2$, $D^2=-D.K_S=2$, $Z_2$ a length $2$ scheme where $D$ fails to be very ample.    
\end{itemize}
\end{prop}





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  in \textit{Grundlehren der Mathematischen Wissenschaften}, Springer
  Verlag, Berlin - Heidelberg - New york - Tokyo (1985).

\bibitem[Ba-So]{ba} E.~Ballico, A.~J.~Sommese,
  \textit{Projective surfaces with $k$-very
    ample line bundles of degree $\leq 4k+4$}, Nagoya Math. J.
  \textbf{136}, 57-79 (1994).

\bibitem[B-F-S]{BFS} M.~Beltrametti, P.~Francia, A.~J.~Sommese,
  \textit{On Reider's method and higher order embeddings}, Duke math. J.
  \textbf{58}, 425-439 (1989).
  
\bibitem[Be-So1]{BS1} M.~Beltrametti, A.~J.~Sommese, \textit{On
    $k$-spannedness 
   for projective surfaces}, in A.~J.~Sommese et. al. (eds.)
   \textit{Hyperplane sections, L'Aquila Proceedings 1988} (Springer LNM 1417,
    pp.~24-51), Berlin, Heidelberg, New York: Springer 1990.

\bibitem[Be-So2]{BS2} M.~Beltrametti, A.~J.~Sommese, \textit{Zero cycles and
    $k$-th order embeddings of smooth projective surfaces}, in
   \textit{Projective surfaces and their classification}, Cortona
   proceedings Symp. Math. INDAM 32, pp.~33-48), New York, London:
   Academic press 1988.

\bibitem[Be-So3]{BS3} M.~Beltrametti, A.~J.~Sommese, \textit{On the preservati%on
   of $k$-very ampleness under adjunction}, Math. Z. \textbf{212},
    257-283 (1993).

\bibitem[Be-So4]{BS4} M.~Beltrametti, A.~J.~Sommese, \textit{The
    Adjunction Theory of Complex Projective Varieties}, De Gruyter
    Expositions in Mathematics \textbf{16}, Berlin: Walter de Gruyter 1995.

%\bibitem[Be-Sz]{BSz} M.~Beltrametti, T.~Szemberg, \textit{On higher order 
%embeddings of Calabi-Yau threefolds}, Arch. Mat. \textbf{74} (2000). 

%\bibitem[Bo]{bog} F.~Bogomolov, \textit{Holomorphic tensors and vector bundles
%on projective varieties}, Math. USSR Izv. \textbf{13}, 499-555 (1979).

%\bibitem[C-F]{CF} F.~Catanese, M.~Franciosi, \textit{Divisors of small genus 
%  on algebraic surfaces and projective embeddings}, in \textit{Proceedings of 
%  the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993)}, 
%  109-140, Israel Math. Conf. Proc. \textbf{9}, Bar-Ilan Univ., Ramat Gan 
%  (1996). 

%\bibitem[C-G]{cg} F.~Catanese, L.~G\"{o}ttsche, \textit{$d$-very ample line 
%bundles and embeddings of Hilbert schemes of $0$-cycles}, Manuscr. Math. 
%\textbf{78}, 193-212 (1990).

\bibitem[C-P]{cp} C.~Ciliberto, G.~Pareschi, \textit{Pencils of minimal 
degree on curves on a $K3$ surface}, J. Reine Angew. Math. \textbf{460}, 
15-36 (1995).

\bibitem[C-M]{cm} M.~Coppens, G.~Martens, \textit{Secant Spaces and 
Clifford's theorem}, Comp. Math. \textbf{68}, 337-341 (1991).

%\bibitem[C-D]{CD} F.~Cossec, I.~Dolgachev, \textit{Enriques Surfaces I}, 
%Birkh{\"a}user 1989.

%\bibitem[Co]{cos} F.~Cossec, \textit{Projective models of Enriques surfaces},
% Math. Ann. \textbf{265}, 283-334 (1983).

\bibitem[De]{dem} M.~Demazure, \textit{Surfaces de Del Pezzo I,II,III,IV,V}, in: M.~Demazure, H.~Pinkham, B.~Teissier (eds.): \textit{Seminaire sur les singularites des surfaces}, Springer LNM \textbf{777} (1980).

\bibitem[DR]{dr} S.~Di Rocco, \textit{$k$-very ample
   line bundles on Del Pezzo surfaces}, Math. Nachr.
   \textbf{179} (1996).

\bibitem[DR2]{dr2} S.~Di Rocco, \textit{Projective surfaces with $k$-very
    ample line bundles of genus $\leq 3k+1$}, Manuscr. Math. \textbf{91},
    35-59 (1996).

 \bibitem[D-M]{dm} R.~Donagi, D.~R.~Morrison, \textit{Linear systems
    on $K3$ sections}, J. Diff. Geom. \textbf{29}, 49-64 (1989).

\bibitem[E-L-M-S]{elms} D.~Eisenbud, H.~Lange, G.~Martens, F.-O.~Schreyer, 
\textit{The Clifford dimension of a projective curve}, Comp. Math. 
\textbf{72}, 173-204 (1989).

\bibitem[G-L]{gl} M.~Green, R.~Lazarsfeld, \textit{Special divisors on
    curves on a $K3$ surface}, Invent. Math. \textbf{89}, 357-370 (1987).

\bibitem[Gr]{gre} M.~Green, \textit{Koszul cohomology and the geometry of 
projective varieties}, J. Differ. Geom. \textbf{19}, 125-171 (1984).

\bibitem[Kn]{kn} A.~L.~Knutsen, \textit{On $k$th order embeddings of $K3$ 
surfaces and Enriques surfaces}, Manuscr. Math. \textbf{104}, 211-237
(2001).

\bibitem[La]{la} R.~Lazarsfeld, \textit{Brill-Noether-Petri without degenerations}, J. Diff. Geom. \textbf{23}, 299-307 (1986).

\bibitem[Ma1]{ma1} G.~Martens, \textit{Funktionen von vorgegebener Ordnung auf 
komplexen Kurven}, J. Reine und Angew. Math. \textbf{320}, 68-85(1980).

\bibitem[Ma2]{ma} G.~Martens, \textit{\"Uber den Clifford-Index algebraischer Kurven}, J. Reine und Angew. Math. \textbf{336}, 83-90 (1982).

\bibitem[M-P]{mp} M.~Mella, M.~Palleschi, \textit{The $k$-very ampleness on 
an elliptic quasi bundle}, Abh. Math. Sem. Univ. Hamburg \textbf{63}, 
215-226 (1993).


\bibitem[Pa]{par} G.~Pareschi, \textit{Exceptional linear systems on curves on Del Pezzo surfaces}, Math. Ann. \textbf{291}, 17-38 (1991).

\bibitem[IRe]{re} I.~Reider, \textit{Vector bundles of rank $2$ and linear systems on algebraic surfaces}, Annals of Math. \textbf{127}, 309-316 (1988). 
 
\bibitem[MRe]{reid} M.~Reid, \textit{Special linear systems on curves lying on a K3 surface}, J. London Math. Soc. (2) \textbf{13}, no. 3, 454-458 (1976).

\bibitem[SD]{S-D} B.~Saint-Donat, \textit{Projective Models of $K3$
  Surfaces}, Amer. J. Math. \textbf{96}, 602-639 (1974).


\bibitem[Se]{ser} F.~Serrano, \textit{Extension of morphisms defined on a 
divisor}, Math. Ann. \textbf{277}, 395-413 (1987).

\bibitem[Te]{ter} H.~Terakawa, \textit{Higher order embeddings of
    algebraic surfaces of Kodaira dimension zero}, Math. Z. \textbf{229},
    417-433 (1998).


\bibitem[Ty]{Ty} A.~N.~Tyurin, \textit{Cycles, curves and vector bundles on
    an algebraic surface}, Duke Math. J. \textbf{54}, 1-26 (1987).

\end{thebibliography}

\end{document}

math0111007.extracted_bib
\begin{thebibliography}{[E-L-M-S]}

\bibitem[ACGH]{acgh} E.~Arbarello, M.~Cornalba, P.~A.~Griffiths,
  J.~Harris, \textit{Geometry of Algebraic Curves, Volume I}, number 267
  in \textit{Grundlehren der Mathematischen Wissenschaften}, Springer
  Verlag, Berlin - Heidelberg - New york - Tokyo (1985).

\bibitem[Ba-So]{ba} E.~Ballico, A.~J.~Sommese,
  \textit{Projective surfaces with $k$-very
    ample line bundles of degree $\leq 4k+4$}, Nagoya Math. J.
  \textbf{136}, 57-79 (1994).

\bibitem[B-F-S]{BFS} M.~Beltrametti, P.~Francia, A.~J.~Sommese,
  \textit{On Reider's method and higher order embeddings}, Duke math. J.
  \textbf{58}, 425-439 (1989).
  
\bibitem[Be-So1]{BS1} M.~Beltrametti, A.~J.~Sommese, \textit{On
    $k$-spannedness 
   for projective surfaces}, in A.~J.~Sommese et. al. (eds.)
   \textit{Hyperplane sections, L'Aquila Proceedings 1988} (Springer LNM 1417,
    pp.~24-51), Berlin, Heidelberg, New York: Springer 1990.

\bibitem[Be-So2]{BS2} M.~Beltrametti, A.~J.~Sommese, \textit{Zero cycles and
    $k$-th order embeddings of smooth projective surfaces}, in
   \textit{Projective surfaces and their classification}, Cortona
   proceedings Symp. Math. INDAM 32, pp.~33-48), New York, London:
   Academic press 1988.

\bibitem[Be-So3]{BS3} M.~Beltrametti, A.~J.~Sommese, \textit{On the preservati%on
   of $k$-very ampleness under adjunction}, Math. Z. \textbf{212},
    257-283 (1993).

\bibitem[Be-So4]{BS4} M.~Beltrametti, A.~J.~Sommese, \textit{The
    Adjunction Theory of Complex Projective Varieties}, De Gruyter
    Expositions in Mathematics \textbf{16}, Berlin: Walter de Gruyter 1995.

%\bibitem[Be-Sz]{BSz} M.~Beltrametti, T.~Szemberg, \textit{On higher order 
%embeddings of Calabi-Yau threefolds}, Arch. Mat. \textbf{74} (2000). 

%\bibitem[Bo]{bog} F.~Bogomolov, \textit{Holomorphic tensors and vector bundles
%on projective varieties}, Math. USSR Izv. \textbf{13}, 499-555 (1979).

%\bibitem[C-F]{CF} F.~Catanese, M.~Franciosi, \textit{Divisors of small genus 
%  on algebraic surfaces and projective embeddings}, in \textit{Proceedings of 
%  the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993)}, 
%  109-140, Israel Math. Conf. Proc. \textbf{9}, Bar-Ilan Univ., Ramat Gan 
%  (1996). 

%\bibitem[C-G]{cg} F.~Catanese, L.~G\"{o}ttsche, \textit{$d$-very ample line 
%bundles and embeddings of Hilbert schemes of $0$-cycles}, Manuscr. Math. 
%\textbf{78}, 193-212 (1990).

\bibitem[C-P]{cp} C.~Ciliberto, G.~Pareschi, \textit{Pencils of minimal 
degree on curves on a $K3$ surface}, J. Reine Angew. Math. \textbf{460}, 
15-36 (1995).

\bibitem[C-M]{cm} M.~Coppens, G.~Martens, \textit{Secant Spaces and 
Clifford's theorem}, Comp. Math. \textbf{68}, 337-341 (1991).

%\bibitem[C-D]{CD} F.~Cossec, I.~Dolgachev, \textit{Enriques Surfaces I}, 
%Birkh{\"a}user 1989.

%\bibitem[Co]{cos} F.~Cossec, \textit{Projective models of Enriques surfaces},
% Math. Ann. \textbf{265}, 283-334 (1983).

\bibitem[De]{dem} M.~Demazure, \textit{Surfaces de Del Pezzo I,II,III,IV,V}, in: M.~Demazure, H.~Pinkham, B.~Teissier (eds.): \textit{Seminaire sur les singularites des surfaces}, Springer LNM \textbf{777} (1980).

\bibitem[DR]{dr} S.~Di Rocco, \textit{$k$-very ample
   line bundles on Del Pezzo surfaces}, Math. Nachr.
   \textbf{179} (1996).

\bibitem[DR2]{dr2} S.~Di Rocco, \textit{Projective surfaces with $k$-very
    ample line bundles of genus $\leq 3k+1$}, Manuscr. Math. \textbf{91},
    35-59 (1996).

 \bibitem[D-M]{dm} R.~Donagi, D.~R.~Morrison, \textit{Linear systems
    on $K3$ sections}, J. Diff. Geom. \textbf{29}, 49-64 (1989).

\bibitem[E-L-M-S]{elms} D.~Eisenbud, H.~Lange, G.~Martens, F.-O.~Schreyer, 
\textit{The Clifford dimension of a projective curve}, Comp. Math. 
\textbf{72}, 173-204 (1989).

\bibitem[G-L]{gl} M.~Green, R.~Lazarsfeld, \textit{Special divisors on
    curves on a $K3$ surface}, Invent. Math. \textbf{89}, 357-370 (1987).

\bibitem[Gr]{gre} M.~Green, \textit{Koszul cohomology and the geometry of 
projective varieties}, J. Differ. Geom. \textbf{19}, 125-171 (1984).

\bibitem[Kn]{kn} A.~L.~Knutsen, \textit{On $k$th order embeddings of $K3$ 
surfaces and Enriques surfaces}, Manuscr. Math. \textbf{104}, 211-237
(2001).

\bibitem[La]{la} R.~Lazarsfeld, \textit{Brill-Noether-Petri without degenerations}, J. Diff. Geom. \textbf{23}, 299-307 (1986).

\bibitem[Ma1]{ma1} G.~Martens, \textit{Funktionen von vorgegebener Ordnung auf 
komplexen Kurven}, J. Reine und Angew. Math. \textbf{320}, 68-85(1980).

\bibitem[Ma2]{ma} G.~Martens, \textit{\"Uber den Clifford-Index algebraischer Kurven}, J. Reine und Angew. Math. \textbf{336}, 83-90 (1982).

\bibitem[M-P]{mp} M.~Mella, M.~Palleschi, \textit{The $k$-very ampleness on 
an elliptic quasi bundle}, Abh. Math. Sem. Univ. Hamburg \textbf{63}, 
215-226 (1993).


\bibitem[Pa]{par} G.~Pareschi, \textit{Exceptional linear systems on curves on Del Pezzo surfaces}, Math. Ann. \textbf{291}, 17-38 (1991).

\bibitem[IRe]{re} I.~Reider, \textit{Vector bundles of rank $2$ and linear systems on algebraic surfaces}, Annals of Math. \textbf{127}, 309-316 (1988). 
 
\bibitem[MRe]{reid} M.~Reid, \textit{Special linear systems on curves lying on a K3 surface}, J. London Math. Soc. (2) \textbf{13}, no. 3, 454-458 (1976).

\bibitem[SD]{S-D} B.~Saint-Donat, \textit{Projective Models of $K3$
  Surfaces}, Amer. J. Math. \textbf{96}, 602-639 (1974).


\bibitem[Se]{ser} F.~Serrano, \textit{Extension of morphisms defined on a 
divisor}, Math. Ann. \textbf{277}, 395-413 (1987).

\bibitem[Te]{ter} H.~Terakawa, \textit{Higher order embeddings of
    algebraic surfaces of Kodaira dimension zero}, Math. Z. \textbf{229},
    417-433 (1998).


\bibitem[Ty]{Ty} A.~N.~Tyurin, \textit{Cycles, curves and vector bundles on
    an algebraic surface}, Duke Math. J. \textbf{54}, 1-26 (1987).

\end{thebibliography}