\section{Introduction}
 \label{s:intro}

 Fake projective planes continue to be among the
most fascinating objects of study in algebraic geometry,
starting from Mumford's construction in the 70's \cite{Mumford}
and culminating in the classification
by Prasad-Yeung \cite{PY} and Cartwright-Steger \cite{CS}.
Here one is solely concerned with smooth projective surfaces
whose (co)homology resembles $\mathbb{P}^2$,
i.e. with Betti numbers $b_2=1, b_1=b_3=0$.

It is then natural to weaken the conditions
by allowing for normal projective surfaces $S$
whose $\Q$-homology equals that of $\mathbb{P}^2$.
Typically, one considers Gorenstein $\Q$-homology projective planes ,
i.e. with only isolated rational double point singularities.
By definition, they are obtained from
smooth projective surfaces by contracting a collection of $(-2)$-curves.
For their understanding it is thus crucial to study the configurations of these curves.

One
 distinguishes whether $K_S$ is ample, numerically trivial or anti-ample.
 The latter case
 corresponds to log del Pezzo surfaces
 and is well understood.
 Indeed, Ye gives a complete classification of  $\Q$-homology log del Pezzo surfaces in \cite{Ye}.
We point out that with one exception they are isolated in moduli.
  On the other end of the scale, the general type $\Q$-homology projective planes
  have resisted a thorough investigation so far.

This papers addresses the case where $K_S$ is numerically trivial.
Here the minimal resolution  is an
 Enriques surface.
 By \cite{HKO}, there are 31 configurations of $(-2)$-curves on Enriques surfaces
 whose contraction may give a $\Q$-homology projective surface,
see Theorem \ref{thm:31}.
  Here the moduli picture has not been determined so far:
   29 of these root types  are supported by examples in \cite{HKO},
(some isolated, some occurring in one-dimensional families),
one further type can be found in \cite{RS}.
This paper's main goal is the full clarification of the picture,
in particular with respect to moduli,
by means of a complete classification of Enriques surfaces
supporting the 31 maximal root types.

\begin{Theorem}
\label{thm}
For the 31 maximal root types realizable on Enriques surfaces,
the following hold
\begin{enumerate}[(i)]
\item
the root types are supported on 1-dimensional families of Enriques surfaces;
\item
the moduli spaces  can have up to 3 different components;
\item
each family has rational base and is defined over $\Q$;
\item
each family can be parametrized explicitly,
see Table \ref{T2}.
\end{enumerate}
\end{Theorem}

The proof of Theorem \ref{thm} crucially relies on the observation
that each Enriques surface arises from a base change type construction
following Kond\=o \cite{Kondo-Enriques} (see also \cite{HS}).
In fact, this is so constructive in nature
that it will allow us to carry out all parametrisations explicitly.

The paper is organized as follows.
Section reviews  root types on Enriques surfaces and the underlying lattice theory
relevant for this paper.
In Section \ref{s:ell}, we relate this to elliptic fibrations
before introducing the main geometric technique
of Enriques involutions of base change type in Section \ref{s:tech}.
Section \ref{s:glue} explains a gluing method
which allows us to compute certain elliptic fibrations
on Enriques surfaces and their jacobians.
In the opposite direction, Section \ref{s:fam}
develops parametrizations on all Enriques surfaces in question
from the information obtained thus far.
This will prove most part of Theorem \ref{thm}.
The final section concerns the number of components of the moduli spaces,
thus completing the proof of Theorem \ref{thm}.
Throughout the paper, we will work out instructive examples in detail
to give a clear idea of the methods and techniques involved.

\subsubsection*{Convention}
We work over $\C$ or any algebraically closed field $K$ of characteristic zero
(although most computations are caried out over $\Q$).
All root lattices $A_n, D_k, E_l$ are taken to be negative-definite.
Orthogonal sums are indicated by a '$+$',
and likewise by $2R$ etc.
The notation $R(m)$ indicates the same abstract $\mathbb{Z}$-module with intersection form multiplied by $m$.

\section{Maximal root types on Enriques surfaces}
\label{s:latt}

By \cite{Ye} (see also the references therein)
there are 27 root lattices supported on
Gorenstein log del Pezzo surfaces of rank 1
(and possibly on surfaces of general type).
This result was complemented in \cite{HKO}
where the case of trivial canonical class was studied.
Necessarily, in this case, the minimal desingularisation
is an Enriques surface $Y$,
and there are 31 possible root types:

\begin{Theorem}[Hwang-Keum-Ohashi]
\label{thm:31}
Let $S$ denote a Gorenstein $\Q$-homology projective plane
with trivial canonical class.
Then the singularity type of $S$ is one of the 31 following:

$A_9$, $A_8 + A_1$, $A_7 + A_2$, $A_7 + 2A_1$,  $A_6 + A_2 + A_1$,   $A_5 + A_4$, $A_5 + A_3 + A_1$, $A_5 + 2A_2$, $A_5 + A_2 + 2A_1$, $2A_4 + A_1$, $A_4 + A_3 + 2A_1$, $3A_3$,  $2A_3 + A_2+ A_1$,
$2A_3 + 3A_1$, $A_3 + 3A_2$,
$D_9$, $D_8 + A_1$, $D_7 + 2A_1$, $D_6 + A_3$, $D_6 + A_2 + A_1$, $D_6 + 3A_1$, $D_5 + A_4$, $D_5 + A_3 + A_1$, $D_5 + D_4$,  $D_4 + A_3 + 2A_1$, $2D_4 + A_1$,
$E_8 + A_1$, $E_7 + A_2$, $E_7 + 2A_1$, $E_6 + A_3$, $E_6 + A_2 + A_1.$
\end{Theorem}

We point out that except for the root types $2A_3 + A_2+ A_1$ and $A_3 + 3A_2$,
each type was supported by an example in \cite{HKO}
-- most of them isolated, but some also occurring in 1-dimensional families.
In parallel, the type $A_3 + 3A_2$ is supported on an isolated example in \cite[Ex. 3.8]{RS}.
Theorem \ref{thm} establishes the full classification of Gorenstein $\Q$-homology projective plane
with trivial canonical class.
Not only does it provide examples supporting the remaining type $2A_3+A_2+A_1$,
but it also confirms the natural conjecture (conf.~Lemma \ref{lem:1-dim})
that each type is supported on one-dimensional families
(thus correcting an erroneous claim for the type $2A_3+3A_1$ from \cite{Keum-Viet}).
We emphasise the contrast with K3 surfaces:
over $\C$, K3 surfaces can support root types of maximal rank 19,
but those surfaces only appear in one-dimensional analytic families
whose algebraic members are isolated, yet countably infinite in number (compare \cite{xiao}).

For later use, we continue to elaborate the lattice theoretic background of Theorem \ref{thm:31}.
Recall that for an Enriques surface $Y$,
the canonical divisor encodes the torsion in $\mathop{\rm Pic}(Y)$.
Quotienting out by numerical equivalence,
we obtain as the free part of $\mathop{\rm Pic}(Y)$ the following unimodular even lattice:
\[
\mathop{\rm Num}(Y) \cong U + E_8
\]
where $U$ denotes the hyperbolic plane.
Assume that $Y$ contains a configuration of nine smooth rational curves
which corresponds to an orthogonal sum $R$ of root lattices of total rank $9$.
Then this gives an embedding
\[
R\hookrightarrow \mathop{\rm Num}(Y)
\]
with orthogonal complement generated by a single positive vector $h$.
More precisely, we find that
$h^2=d>0$ where $-d$ is the discriminant of the primitive closure $R'$ of $R$ in $\mathop{\rm Num}(Y)$:
\[
R' = (R\otimes\Q)\cap\mathop{\rm Num}(Y).
\]
Following Nikulin \cite{Nikulin}
$R'$ and $\mathbb{Z} h$ glue together to the unimodular even lattice $\mathop{\rm Num}(Y)$ by means of an isomorphism of discriminant groups
\[
A_{R'} = (R')^\vee/R' \cong A_{\mathbb{Z} h} \cong \mathbb{Z}/d\mathbb{Z},
\]
which extends to an isomorphism of the discriminant forms modulo $2\mathbb{Z}$
\[
q_{R'}\cong -q_{\mathbb{Z} h}.
\]
In the present situation, this details as follows:

\begin{Lemma}
\label{lem:R}
$R$ may arise from a configuration of smooth rational curves on an Enriques surface
if it admits an even finite index overlattice $R'$ such that
\[
q_{R'} \cong \mathbb{Z}/d\mathbb{Z}(-1/d).
\]
\end{Lemma}

\begin{Remark}
Theorem \ref{thm} implies that the condition from Lemma \ref{lem:R} is not only necessary,
but also sufficient as soon as $R$ has at most 4 orthogonal summands or $R=2A_3+3A_1$.
\end{Remark}

Note that  the even finite index overlattices of $R$ are encoded in the discriminant form $q_R$ as well.
Hence we can retrieve all information necessary to check the condition of Lemma \ref{lem:R}
from the discriminant form $q_R$.
In \cite{HKO}, the authors decided to rather argue with local epsilon-invariants,
but for our purposes it will be more beneficial to work with discriminant forms.
Before illustrating this with three instructive examples,
we fix notation for discriminant forms of root lattices.
All of this is well-known;
for instance, it is encoded in the correction terms of the height pairing
on Mordell-Weil lattices \cite{ShMW}.

\subsection{Root lattice $A_n \; (n>0)$}

We number the vertices of the corresponding Dynkin diagram $a_1,\hdots,a_n$.

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(80,10)(0,25)
\multiput(3,32)(20,0){5}{\circle*{1.5}}
\put(3,32){\line(1,0){80}}
\put(2,25){$a_1$}
\put(22,25){$a_2$}
\put(50,25){$\hdots$}
\put(82,25){$a_n$}

\end{picture}
\end{figure}

Then each vertex $a_i$ defines a unique dual vector
\[
a_i^\vee\in A_n^\vee \;\;\; \text{ such that } \;\;\; a_i^\vee.a_j = \delta_{ij}.
\]
Here
\[
(a_i^\vee)^2 = -\frac{i(n+1-i)}{n+1} \;\;\; \text{ and } \;\;\; A_n^\vee/A_n = \langle a_1^\vee\rangle \cong \mathbb{Z}/(n+1)\mathbb{Z}
\]
with the property that $a_m^\vee \equiv m\cdot a_1^\vee \mod A_n$.

\subsection{Root lattice $D_k \; (k\geq 4)$}

We number the vertices of the corresponding Dynkin diagram $d_1,\hdots,d_k$.

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(100,20)(5,-5)
\multiput(3,8)(20,0){5}{\circle*{1.5}}
\put(3,8){\line(1,0){80}}
\put(83,8){\line(2,1){17}}
\put(83,8){\line(2,-1){17}}
\put(100,16){\circle*{1.5}}
\put(100, 0){\circle*{1.5}}
\put(2,1){$d_1$}
\put(22,1){$d_2$}
\put(78,1){$d_{k-2}$}
\put(103,16){$d_{k-1}$}
\put(103, 0){$d_{k}$}

\end{picture}
\end{figure}

Then each vertex $d_i$ defines a unique dual vector
\[
d_i^\vee\in D_k^\vee \;\;\; \text{ such that } \;\;\; d_i^\vee.d_j = \delta_{ij}.
\]
Here
\[
(d_1^\vee)^2 = -1   \;\;\; \text{ and } \;\;\; (d_{k-1}^\vee)^2 = (d_k^\vee)^2 = -\frac k4
\]
with $d_1^\vee.d_{k-1}^\vee = d_1^\vee.d_{k}^\vee = -1/2$
and
$d_{k-1}^\vee.d_{k}^\vee = -(k-2)/4$.
One has
\[
D_k^\vee / D_k =
\begin{cases}
\langle d_k^\vee\rangle \cong \mathbb{Z}/4\mathbb{Z} & \text{ if $k$ is odd,}\\
\langle d_1^\vee, d_k^\vee\rangle \cong (\mathbb{Z}/2\mathbb{Z})^2 & \text{ if $k$ is even.}
\end{cases}
\]

\subsection{Root lattice $E_l \; (l=6,7,8)$}
\label{ss:E_l}

We number the vertices of the corresponding Dynkin diagram $e_1,\hdots,e_l$.

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(120,30)(3,-40)
\multiput(3,-32)(20,0){7}{\circle*{1.5}}
\put(3,-32){\line(1,0){120}}
\put(2,-39){$e_2$}
\put(22,-39){$e_3$}
\put(42,-39){$e_4$}
\put(62,-39){$e_5$}
\put(43,-32){\line(0,1){20}}
\put(43,-12){\circle*{1.5}}
\put(46,-13){$e_1$}
\put(92,-39){$\hdots$}
\put(122,-39){$e_l$}

\end{picture}
\end{figure}

There is a unique dual vector
\[
e_l^\vee\in E_l^\vee \;\;\; \text{ such that } \;\;\; e_l^\vee.e_j = \delta_{lj}.
\]
Here
\[
(e_l^\vee)^2 = -\frac{10-l}{9-l} \;\;\; \text{ and } \;\;\; E_l^\vee/E_l = \langle e_l^\vee\rangle\cong \mathbb{Z}/(9-l)\mathbb{Z}.
\]

\begin{Example}[$R=A_9$]
\label{ex:1}
Take the root lattice $R=A_9$ (the first from the list in Theorem \ref{thm:31}).
Since $R$ has square-free discriminant $-10$,
there are no integral overlattices.
Hence we have to check whether
\[
A_9 + \langle 10\rangle
\]
glue together to the even unimodular lattice $U+E_8$.
The generators of the discriminant groups have square
$(a_1^\vee)^2=-9/10$ and $1/10$.
Since these agree up to sign and a square factor,
the gluing is easily achieved.
\end{Example}

\begin{Example}[$R=A_8+A_1$]
\label{ex:2}
Take the root lattice $R=A_8+A_1$ (the second from the list in Theorem \ref{thm:31}).
The only integral overlattice is obtained by adjoining the element $a_3^\vee\in A_8^\vee$ to $A_8$.
Necessarily this leads to the unimodular lattice $E_8$.
We discuss two cases:

If $R'=E_8+A_1$, then $h^2=2$ and the gluing of $A_1$ and $\mathbb{Z} h$ to $U$ is immediate.

If $R=R'=A_8+A_1$, then we require $h^2=18$.
A generator of $A_R$ is given by the sum of both $a_1^\vee$'s for $A_8$ and $A_1$.
It has square $-8/9-1/2=-25/18$ which agrees up to sign and a square with $(h/18)^2=1/18$.

In conclusion, according to Lemma \ref{lem:R}, both cases may a priori occur on an Enriques surfaces.
We will see in \ref{ss2:A_8+A_1} that this indeed gives rise to two distinct cases in practice,
but in an unexpected way.
\end{Example}

\begin{Example}[$R=A_7+A_2$]
This is the first case from Theorem \ref{thm:31}
where we can rule out some gluing
since the discriminant forms do not match.
Namely, if $R=R'$,
then we would need to glue $R$ with a vector $h$ of square $h^2=24$.
However, the generators of the $8$-torsion in the discriminant groups,
$a_1^\vee\in A_7^\vee$ and $h/8$,
evaluate as
\[
{a_1^\vee}^2 = -\frac 78,\;\;\; (h/8)^2 = \frac 38.
\]
Since these are not compatible modulo $2\mathbb{Z}$ up to sign,
$A_7+A_2$ cannot be embedded primitively into the even unimodular lattice $U+E_8$.

Consider instead $R'=E_7+A_2$.
Here the gluing with a vector $h$ of square $h^2=6$ is easily achieved
using $e_7^\vee$ and $h/2$ or $a_1^\vee$ and $h/3$.
\end{Example}

\section{Elliptic fibrations on Enriques surfaces}
\label{s:ell}

It is one of the key features of Enriques surfaces $Y$
that they always admit elliptic fibrations
\begin{eqnarray}
\label{eq:ell}
f:\; Y \to \mathbb{P}^1,
\end{eqnarray}
and in fact often many.
Necessarily they come with two fibers of multiplicity two
(whose supports are usually called half-pencils)
and with some bisection
(so strictly speaking, the generic fiber is a genus one curve,
but not endowed with a rational point over $K(\mathbb{P}^1)$).
There are (at least) two ways to see this.
On the one hand,
consider the universal covering $X$ which is a K3 surface:
\[
\pi: X \to Y
\]
Naturally the elliptic fibration carries over to $X$,
and it is compatible with the fixed-point free involution $\imath$
whose quotient recovers $Y$.
Here $\imath$ also acts on the base $\mathbb{P}^1$ of the fibration,
and the multiple fibers of $Y$ sit over the fixed points.

Alternatively,
consider the jacobian fibration
\[
\mathop{\rm Jac}(Y) \to \mathbb{P}^1.
\]
This is a rational elliptic surface
with the same singular fibers as $Y$,
but of course with a section.
Reversely, starting from $\mathop{\rm Jac}(Y)$, one can head in two different directions.
First, the original surface $Y$
can be recovered analytically by a logarithmic transformation.
Secondly, one can apply a quadratic base change ramified in the two fixed point of $\imath$ on $\mathbb{P}^1$.
This gives rise to an elliptic K3 surface with section --  the jacobian of $X$.

What makes elliptic fibrations on K3 and Enriques surfaces so useful
is their compatibility with lattice theory.
Most prominently, any non-zero divisor $D$
on a K3 surface $X$ of square $D^2=0$ gives rise to an elliptic fibration
after subtracting the base locus from $D$ or $-D$, whichever is effective
(cf.~\cite{PSS}),
and any root lattice $R\subset\mathop{\rm NS}(X)$ perpendicular to $D$ corresponds to fiber components
(compare \cite{Kondo-Aut} and Prop.~\ref{prop:E}, Lem.~\ref{prop:fibers}).
On Enriques surfaces,
a similar relation persists,
but with a striking difference:
on a K3 surface $X$, for any divisor $D$ of square $D^2=-2$, either $D$ or $-D$ is effective by Riemann-Roch.
Consequently it supports a connected configuration of smooth rational curves,
and root lattices inside $\mathop{\rm NS}(X)$ come from smooth rational curves.
On an Enriques surface $Y$, however, this is far from being true.
This can already be seen in the decomposition $\mathop{\rm Num}(Y)=U+E_8$:
clearly $E_8$ is supported on classes of square $-2$,
but a general Enriques surface contains no smooth rational curves at all \cite{BP}.

\begin{Example}[$R=A_8+A_1$ cont'd]
\label{ex:2-2}
For the root type $R=A_8+A_1$,
we distinguished two cases in Example \ref{ex:2}.
If $R$ is primitive in $\mathop{\rm Num}(Y)$, i.e. $R=R'$,
then it is supported on smooth rational curves by assumption.
If $R$ is not primitive, i.e. $R'=E_8+A_1$,
then we have to distinguish further whether $R'$, or equivalently $E_8$,
is supported on $(-2)$-curves or not.
Once we verify that both cases occur (and are genuinely distinct),
it will follow that one component of the moduli space for Enriques surfaces with root type $R$
is in fact given by the one for root type $R'$.
This subtlety will play an important role throughout this paper, see also \ref{ss:A_8+A_1} and  especially Section \ref{s:number}.
\end{Example}

\begin{Definition}
Let $Y$ be an Enriques surface
and $R\subset\mathop{\rm Num}(Y)$  a root lattice supported on $(-2)$-curves
on $Y$.
We call $R$ \emph{nodally primitive}
if it has no finite index overlattice in $\mathop{\rm Num}(Y)$ supported on $(-2)$-curves.
\end{Definition}

In the opposite direction, consider an Enriques surface containing a smooth rational curve $C$.
It is immediate that $\pi^*C$ splits into two disjoint effective divisors of square $-2$ on $X$.
That is, by the above discussion, smooth rational curves on Enriques surfaces are duplicated on the K3 cover.
We directly obtain the following corollary
which shows that Theorem \ref{thm} is the best we can hope for
if it comes to moduli dimensions:

\begin{Lemma}
\label{lem:1-dim}
Enriques surfaces with root type $R$ of rank $9$ come in 1-dimensional families at most.
\end{Lemma}

\begin{proof}
Moduli of Enriques surfaces are governed by the covering K3 surfaces.
Denote the Enriques surface with root type $R$ of rank $9$ by $Y$
and the K3 cover by $X$.
By what we have said before,
we have an embedding
\[
2R \hookrightarrow \mathop{\rm NS}(X).
\]
On the other hand, $X$ is automatically algebraic as there is another embedding
\[
U(2)+E_8(2) \cong \pi^*\mathop{\rm Num}(Y) \hookrightarrow \mathop{\rm NS}(X).
\]
Since $R^2$ is negative-definite of rank $18$ while $\mathop{\rm Num}(Y)$ is hyperbolic,
we find that $\rho(X)\geq 19$.
It is known from the theory of lattice polarised K3 surfaces,
that such K3 surfaces come in 1-dimensional families.
\end{proof}

We are now in the position to record the properties of elliptic fibrations
on Enriques surfaces relevant for our purposes.
Beforehand, we point out another important difference to K3 surfaces
next to the subtleties involving $(-2)$-classes:
for a half-pencil $E$, only $|2E|$ will induce an elliptic fibration, but not $|E|$.

\begin{Proposition}
\label{prop:E}
Let $Y$ be an Enriques surface
and $E$ a divisor with non-zero class in $\mathop{\rm Num}(Y)$ and square $E^2=0$.
Then
\begin{enumerate}[(i)]
\item
$E$ gives rise to an elliptic fibration on $Y$;
\item
any smooth rational curve $C$ perpendicular to $E$
gives a fiber component of the fibration;
\item
in particular, any root lattice $R\subset\mathop{\rm Num}(Y)$
which is perpendicular to $E$ and
supported on $(-2)$-curves on $Y$,
embeds into the singular fibers of the fibration;
\item
if $Y$ contains a smooth rational curve $B$ with $B.E=1$,
then $E$ gives a half-pencil, and $B$ gives a bisection.
\end{enumerate}
\end{Proposition}

\begin{proof}
\textit{(i)}
By Riemann-Roch, either $E$ or $-E$ is effective,
say w.l.o.g $E$.
If its class in $\mathop{\rm Num}(Y)$ is not primitive,
then divide by the degree of primitivity
and continue with the primitive class thus obtained.
By \cite[(17.4)]{BHPV},
it remains to subtract the base locus of $|2E|$;
this is given by trees of $(-2)$-curves.
Hence it suffices to apply successive reflections in these curves
to arrive at a half-pencil.

\textit{(ii), (iii)}
We argue with the procedure sketched above to derive a half-pencil from $E$.
The only critical step consists in the reflections
since these generally do not act effectively on $\mathop{\rm Num}(Y)$.
On the classes of $(-2)$-curves, however, they do act effectively up to sign.
This follows from the fact that the $(-2)$-curves split on the K3 cover $X$,
and likewise that a reflection on $Y$ corresponds to a composition of reflections on $X$
(compare the discussion in \cite[\S 2.3]{RS}).
Thus we can assume that the smooth rational curve $C$, or the root lattice $R$ supported on $(-2)$-curves on $Y$,
is perpendicular to the half-pencil $E$.
That is, their classes embed into
$E^\perp/\mathbb{Z} E$,
and we conclude as in \cite[Lem. 2.2]{Kondo-Aut}
that the $(-2)$-curves correspond to fiber components
(but unlike loc. cit. we do not claim that they determine the singular fibers,
since we have no assumption such as $E^\perp/\mathbb{Z} E=R$, see Lemma \ref{prop:fibers}).

\textit{(iv)}
By inspection of the intersection number, $E$ has primitive class in $\mathop{\rm Num}(Y)$.
We continue as in \textit{(i)}, using the special action of the reflections on $(-2)$-curves
as laid out in \textit{(ii), (iii)}.
If $E$ is effective, then  the reflections yield a $(-2)$-curve $B'$
meeting the half-pencil $E'$ induced by $E$ with multiplicity one.
If $-E$ is effective, then $B$ is contained in the support of $-E$.
The reflection in $B$ gives an effective isotropic class $E'=-E-B$
such that $B.E'=1$. Now continue as before to complete the proof of Proposition \ref{prop:E}.
\end{proof}

We emphasize the most instructive case of Proposition \ref{prop:base}:
when $E$ itself happens to be a divisor of Kodaira type,
then it naturally appears as a singular fiber of the induced fibration.
For later use, we continue to elaborate on the relation between the root lattice $R$ in Proposition \ref{prop:E}  \textit{(iii)}
and the underlying singular fibers.
To this end, we recall that the fibers correspond to the extended Dynkin diagrams $\tilde A_n, \tilde D_k, \tilde E_l$.

\begin{Lemma}
\label{prop:fibers}
Let $E$ be a non-trivial isotropic vector in $\mathop{\rm Num}(Y)$
and $R$ a root lattice supported on $(-2)$-curves on $Y$.
Let $R_0= E^\perp \cap R$ decompose into irreducible root lattices
\[
R_0 = \sum_v R_v.
\]
Then the following are equivalent:
\begin{enumerate}
\item
 the singular fibers of the elliptic fibration induced by $E$
support a root lattice strictly greater than $R_0$;
\item
the reducible fibers are not in bijective correspondence to the extended Dynkin diagrams $\tilde R_v$;
\item
$R_0$ is a proper sublattice of the span of the $(-2)${-curves} inside $E^\perp/\mathbb{Z} E$.
(i.e.  $R_0$ it is not nodally primitive or not of finite index  in there).
\end{enumerate}
\end{Lemma}

\begin{proof}
The equivalence of \text{(1), (2)} follows from Kodaira's classification:
$\tilde R_v$ gives the singular fiber $F_v$ if and only if in $R_v$ a simple fiber component is omitted (cf.~Rem.~\ref{rem:unique}).
Otherwise $R_v$ will have finite index in the root lattice corresponding to the fiber $F_v$.
The same applies to a sum of irreducible root lattices embedding into a single fiber.

\textit{(3)} is exactly how Kond\=o's argument in \cite[Lem. 2.2]{Kondo-Aut} has to be rephrased
if  $E^\perp/\mathbb{Z} E$ is not a priori a root lattice (both on Enriques and K3 surfaces):
the singular fibers are encoded in the root lattice inside $E^\perp/\mathbb{Z} E$
generated by  $(-2)$-curves on $Y$ perpendicular to $E$.
Hence the equivalence with \textit{(1), (2)} follows.
\end{proof}

The next corollary draws the consequence for our situation
of a maximal root type.
The arguments can easily be adapted for the general case
which we omit for shortness.

\begin{Corollary}
\label{cor:fibers}
Let $Y$ be an Enriques surface with root type $R$ of rank $9$
supported on $(-2)$-curves.
Assume that there is a non-trivial isotropic class $E\in\mathop{\rm Num}(Y)$
such that $R_0=E^\perp\cap R$ is a root lattice of rank $8$.
If $R$ is nodally primitive,
then the reducible singular fibers of the elliptic fibration induced by $E$
are given by $\tilde R_v$ for the irreducible orthogonal summands $R_v$ of $R_0$.
\end{Corollary}

\begin{proof}
$R_0$ has finite index in $E^\perp/\mathbb{Z} E$ since both lattices are negative-definite of rank $8$.
Assuming that the reducible  fibers are not all given by the $\tilde R_v$,
we deduce from Lemma \ref{prop:fibers} that $R_0$ is not nodally primitive.
On the other hand, by the very definition, $R_0=E^\perp\cap R$
is primitively embedded into $R$.
Hence $R$ cannot be nodally primitive either.
\end{proof}

\begin{Remark}
The above implication is not an equivalence,
i.e.~$R_0$ may be nodally primitive without $R$ being so,
see Remark \ref{rem:prim}.
Of course, this weaker assumption suffices to conclude as in Corollary \ref{cor:fibers}
that the singular fibers are determined by $R_0$.
\end{Remark}

In the next section,
we start translating the lattice-theoretic information from this section
into explicit equations of the elliptic fibrations.
For this purpose, it will be instrumental to
determine how the bisection intersects the singular fibers.
The subtlety in this respect lies in the fact
that we have to consider the nodal primitive closure.
We only phrase the criterion for the special case relevant to our considerations:

\begin{Lemma}
\label{lem:bi}
In the set-up of Corollary \ref{cor:fibers},
assume that the support of the root lattice $R$
also contains the class of an smooth rational curve $B$
such that $B.E=1$.
If $R_0$ is nodally primitive,
then the  bisection of the elliptic fibration arising from $B$
meets the reducible singular fibers exactly in the components predicted by $R_v$.
\end{Lemma}

Indeed, as discussed above, it is exactly when $R_0$ is nodally primitive
that the underlying singular fibers of type $\tilde R_v$ are obtained
by supplementing $R_v$ with a simple fiber component.
The latter is met by $B$ with multiplicity complementary to the intersection number with $R_v$
(taking into account the multiplicity of the fiber).
We emphasis that if $R_0$ is not nodally primitive,
then the nodal primitive closure gives the singular fibers.
Hence, as negative-definite lattices, there only is a finite-index difference,
but in terms of $(-2)$-curves supported on the singular fibers,
the closure adds a multiple fiber component to the $R_v$'s
which ensures that all fiber components are indeed present.
In particular, they cannot all be perpendicular to the bisection.
The following example will illustrate this quite clearly.

\begin{Example}[$R=A_8+A_1$ again]
\label{ex:2-3}
Take $R=A_8+A_1$.
Following up on Example \ref{ex:2-2},
consider the imprimitive case where $R'=E_8+A_1$.
The gluing from Example \ref{ex:2}
gives rise to the isotropic vector $E=a_1^\vee+h/2$;
by Proposition \ref{prop:E} the $(-2)$-curve $B$ generating the $A_1$-summand gives a bisection
of the  elliptic fibration arising from $E$.
We have $R_0=A_8$ and shall distinguish whether $R_0$ is nodally primitive or not.
If so, then $R_0$ gives a singular fiber of type $I_9$
where the bisection $B$ only meets a single fiber component by Lemma \ref{lem:bi}.
In contrast, if $R_0$ is not nodally primitive,
but the $(-2)$-curves really support the root lattice $E_8$,
then the corresponding singular fiber has type $II^*$.
Again, by Lemma \ref{lem:bi}, the bisection $B$ only meets a single fiber component (with multiplicity two),
but it is the the simple one, say $e_9$, off $E_8$.
In particular,  $B$ is not perpendicular to the root lattice $A_8$ embedded into the singular fibers as $\langle e_2,\hdots,e_9\rangle$.
\end{Example}

We end this section by pointing out that these subtle relations between $(-2)$-classes and curves,
and between singular fibers and root lattices
are the main reason why it is highly non-trivial to decide
whether a root lattice $R$ from Theorem \ref{thm:31} is supported on an Enriques surface
for some proper overlattice of $R$.
For the details, see Section \ref{s:number}.

\section{Enriques involutions of base change type}
\label{s:tech}

This section introduces the main theoretical ingredient
for the proof of Theorem \ref{thm}.
We review a construction of Enriques involutions from \cite{HS}
based on quadratic twists
which can be controlled very well and is nicely compatible with
the theories of lattice polarised K3 surfaces
and Mordell-Weil lattices.

The overall set-up is as follows:
assume that an elliptic fibration \eqref{eq:ell} on an Enriques surface $Y$
admits a bisection $C$
which splits on the K3 cover $X$ into two smooth rational curves
(not necessarily disjoint).
The induced elliptic fibration on $X$ is thus equipped with two sections
one of which, say $O$, we choose as zero for the group law on the fibers.
In particular,
$X=\mathop{\rm Jac}(X)$ and we have a commutative diagram consisting of elliptic fibrations
and $2:1$ maps:
$$
\begin{array}{ccccl}
&& X &&\\
& \swarrow &  \downarrow & \searrow &\\
Y && \mathbb{P}^1 && S=\mathop{\rm Jac}(Y)\\
\downarrow & \swarrow && \searrow & \downarrow\\
\mathbb{P}^1 & \multicolumn{3}{c} = & \mathbb{P}^1
\end{array}
$$
Let $\jmath$ denote the deck transformation corresponding to the double covering $X\to S$.
Then one can show that the other section mapping to $C$,
say $P$, is anti-invariant in $\mathop{\rm MW}(X)$ for $\jmath^*$,
and that the Enriques involution $\imath$ equals $\jmath$ composed with translation by $P$
\cite[Prop.~3.1]{HS}.

The crucial point of this construction is that
the shape of $P$ can be predicted precisely,
so that often $P$ (and everything else) can be computed explicitly.
Notably, $P$ is invariant for the composition of $\jmath$ and $-1$ (acting fiberwise),
hence it descends to the quotient of $X$ by $\jmath\circ(-1)$.
The latter automorphism of $X$ is a Nikulin involution with four fixed points  in each fiber fixed by $\jmath$,
so the minimal desingularisation of $X/(\jmath\circ(-1))$ is again K3.
In fact, this is exactly the quadratic twist of $S$ at the two fixed point of $\jmath$ (or $\imath$)
acting on $\mathbb{P}^1$.
We denote the quadratic twist by $S'$.
Explicitly,
if $S$ is given in extended Weierstrass form
\begin{eqnarray}
\label{eq:S}
S:\;\;\; y^2 = f(x),\;\;\; f\in K(t)[x],\; \;\; \deg f=3,
\end{eqnarray}
then the quadratic twist at the zeroes of a quadratic polynomial $q\in K(t)[x]$ (possibly linear, i.e.~including a zero at $\infty$)
is given by
\begin{eqnarray}
\label{eq:S'}
S': \;\;\; q(x)y^2 = f(x).
\end{eqnarray}
Here both $S$ and $S'$ pull-back to $X$ under the degree two morphism of $\mathbb{P}^1$ ramified at the roots of $q(x)$,
and the section $P$ is induced by a section $P'\in\mathop{\rm MW}(S')$.
Basically, the only condition for the composition of $\jmath$ and translation by $P$
to be fixed point-free is that $P'$ meets both ramified fibers on $S'$
in a component different from the identity component (the one met by the zero section
which is chosen compatibly with $O$).

While the above technique is extremely useful
for explicit computations and constructions,
it is in fact not so easy to detect on the level of Enriques surfaces
when it applies.
That is, of course, unless the bisection $C$ itself is smooth rational
because then it splits automatically on $X$ as discussed in Section \ref{s:ell}.

\begin{Lemma}
\label{lem:bct}
Let $Y$ be an Enriques surface with given elliptic fibration admitting a smooth rational bisection.
Then $Y$ arises from an Enriques involution of base change type
originating from $\mathop{\rm Jac}(Y)$.
\end{Lemma}

This basic case
which originally goes back to Kond\=o \cite{Kondo-Enriques}
before being generalized in \cite{HS}
will be instrumental for our paper,
so we continue to elaborate on the details.

Assuming that the bisection $C$ is smooth rational as in Lemma \ref{lem:bct} (i.e. $C^2=-2$),
it splits on $X$ into orthogonal sections $O, P$.
In other terminology, $P$ is an \emph{integral} section (relative to the neutral element $O$).
One can easily translate this into the shape of $P'$.
To this end, we make sure that the extended Weierstrass forms involved are minimal.
In terms of \eqref{eq:S}, \eqref{eq:S'},
one can assume after some variable transformations that
\[
f = x^3 + a_2 x^2 + a_4 x + a_6, \;\;\; a_i\in K[t].
\]
Then the minimality of \eqref{eq:S} is equivalent to the degree condition
\[
\deg a_i \leq i \;\;\; (i=2,4,6) \;\;\; \text{ with some } a_i\not\in K \text{ and  some } a_j/t^j\not\in K.
\]
In consequence, the integral sections of $S$ are given by pairs of polynomials
$(U,V)$ in $t$ such that $\deg U\leq 2, \deg V\leq 3$.
In what follows,
we will abbreviate these polynomials (or in general rational functions)
by $x(P), y(P)$.

Accordingly, on $S'$ we shall concentrate on integral sections $P'$
meeting both ramified fibers in non-identity components.
In terms of \eqref{eq:S'},
these conditions translate as
\begin{eqnarray}
\label{eq:x(P')}
x(P'), y(P') \in K[t] \;\;\; \text{ with } \;\;\;
\deg x(P'), \deg y(P') \leq 2.
\end{eqnarray}
In practice,
given $S$ and $q$,
this approach often leads to a system
of polynomial equations
which can be solved directly or with a computer algebra system.
It turns out to be even more beneficial
for computing families of Enriques surfaces and their covering K3 surfaces
as then we allow $q$ to vary as well.
An example in this spirit was given for the generic nodal Enriques surface in \cite[3.8]{HS}.
Here we illustrate this with a 1-dimensional family geared towards our aims.

\begin{Example}[$R=A_9$ cont'd]
\label{ex:1-2}
Consider one of the rational elliptic surface with finite Mordell-Weil group $\mathop{\rm MW}=\mathbb{Z}/4\mathbb{Z}$:
\[
X_{8211}: \;\;\; y^2 = x(x^2+(t^2+2)x+1).
\]
This has singular fibers of Kodaira types $I_8$ at $\infty, I_2$ at $0$ and $I_1$ at the roots of $t^2+4$.
There are 4-torsion sections $(-1,\pm t)$ and a 2-torsion section $(0,0)$.
For any quadratic twist, the 2-torsion section is simultaneously invariant and anti-invariant  for the deck transformation $\jmath^*$,
so it leads to an Enriques involution if and only if there is no ramification at the $I_2$ fiber.
This leads to a 2-dimensional family of Enriques surfaces with root types, e.g., $E_7+A_1$ and $A_7+A_1$,
but by Lemma \ref{lem:1-dim}, it cannot  support as a whole any of the types from Theorem \ref{thm:31}.

In order to support Enriques surfaces of root type $A_9$,
we arrange for the quadratic twist at $q$
to admit a section $P'$ meeting both $I_8$ and $I_2$ fibers at non-identity components
adjacent to the identity component.
This translates as
\[
x(P')|_{t=0} = -1,\;\;\; x(P')|_{t=\infty} = 0.
\]
By \eqref{eq:x(P')}, this directly leads to
\[
x(P') = \mu t -1
\]
which lives on the quadratic twist at
\[
q= (\mu t-1) (\mu t +\mu^2-1).
\]
The base change type technique thus gives a one-dimensional family of Enriques surfaces
(properly parametrised by $\sqrt \mu$ after eliminating symmetries)
with reducible singular fibers of Kodaira types $I_8, I_2$ and a smooth rational bisection $C$
(the image of $O$ and $P$).
The Enriques surfaces (for $\mu\neq 0$) feature the following diagram of $(-2)$-curves
which obviously support the root type $A_9$:

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(100,20)(0,0)
\multiput(0,0)(20,0){4}{\circle*{1.5}}
\put(0,0){\line(1,0){60}}

\multiput(0,20)(20,0){4}{\circle*{1.5}}
\put(0,20){\line(1,0){60}}

\put(0,20){\line(0,-1){20}}
\put(60,20){\line(0,-1){20}}

\put(60,20){\line(2,-1){40}}
\put(60,0){\line(2,1){40}}

\put(99,20){\line(0,-1){20}}
\put(101,20){\line(0,-1){20}}

\put(80,10){\circle*{1.5}}
\put(77,2){$C$}

\put(100,20){\circle*{1.5}}
\put(100,0){\circle*{1.5}}

\end{picture}
\caption{$(-2)$-curves supporting the root type $A_9$}
\label{fig:A_9}
\end{figure}
\end{Example}

In the next section we shall see how all Enriques surfaces
supporting root types of rank $9$ (thus from Theorem \ref{thm:31})
arise naturally through the base change construction,
and in fact from very special rational elliptic surfaces (see Proposition \ref{prop:base}).

\section{Gluing technique}
\label{s:glue}

The aim of this section is to prove a  classification result which
will be crucial for the proof of Theorem \ref{thm}.
In essence, we narrow down the rational elliptic surfaces
required to study
Enriques surfaces
supporting root types of rank $9$
to the following very special class:

\begin{Definition}
\label{def}
A rational elliptic surface $S\to \mathbb{P}^1$ is called \emph{extremal}
if its Mordell-Weil group $\mathop{\rm MW}(S)$ is finite.
\end{Definition}

Extremality translates into singular fibers and lattices as follows:

\begin{Criterion}
\label{crit}
A rational elliptic surface is extremal if and only if
the singular fibers support  root lattices of total rank $8$.
\end{Criterion}

\begin{proof}
Following Kodaira's classification \cite{K},
the singular fibers are associated with root lattices $R_v$
of rank one less than the number of fiber components
(compare Lemma \ref{prop:fibers}).
Writing $R=\oplus_v R_v$,
the Shioda-Tate formula reads
\begin{eqnarray}
\label{eq:ST}
\rho(S) = 2 + \mathop {\rm rk} R + \mathop {\rm rk} \mathop{\rm MW}(S).
\end{eqnarray}
Recall that any rational elliptic surface $S$ is isomorphic to $\mathbb{P}^2(9)$,
the projective plane blown up in the base points of a cubic pencil.
In particular, $\rho(S)=10$,
and finite $\mathop{\rm MW}$ is equivalent to $\mathop {\rm rk} R = 8$ by \eqref{eq:ST} as claimed.
\end{proof}

\begin{Remark}
\label{rem:unique}
In associating a root lattice $R_v$ to the singular fiber at $v$,
one usually omits the fiber component met by the zero section.
This choice is particularly compatible with the theory of Mordell-Weil lattices \cite{ShMW},
but for additive fibers, it is by no means unique
as we have seen in Section \ref{s:ell}, especially in Lemma \ref{prop:fibers}.
\end{Remark}

We point out that there are only 16 extremal rational elliptic surfaces,
15 of which isolated
while one occurs in a one-dimensional continuous family (varying with the j-invariant).
They were classified by Miranda-Persson in \cite{MP}.
We will give all the relevant equations and properties in Table \ref{T0} below,
but in a different form geared towards our purposes.
We are now in the position to state the key result for this section:

\begin{Proposition}
\label{prop:base}
Any Enriques surface supporting a root type of rank $9$
arises from an extremal rational elliptic surface
by the base change type construction.
\end{Proposition}

The proof of Proposition \ref{prop:base} will cover the remainder of this section.
In the end, we will be rather sketchy,
but to give an idea of the concepts and obstacles involved
we will work out a few instructive cases in full detail.
The key idea is as follows:
using Proposition \ref{prop:E}
exhibit an elliptic fibration on the Enriques surface $Y$
whose singular fibers support root lattices of total rank $8$.
Since $\mathop{\rm Jac}(Y)$ shares the  singular fibers with $Y$,
Criterion \ref{crit} shows
that $\mathop{\rm Jac}(Y)$ is an extremal rational elliptic surface
(which we often determine explicitly for later use in the proof of Theorem \ref{thm}).
Then it remains to equip $Y$ with a smooth rational bisection
and apply Lemma \ref{lem:bct}.

\subsection{Proof of Proposition \ref{prop:base} for $\boldsymbol{R=A_9}$}
\label{ss:A_9}

Let $Y$ be an Enriques surface supporting the root type $A_9$.
Recall from Example \ref{ex:1} that we have an orthogonal sum
\[
A_9 + \mathbb{Z} h \subset \mathop{\rm Num}(Y), \;\;\; h^2=10.
\]
We shall apply Proposition \ref{prop:E} to the isotropic vector
\[
E=a_2^\vee + 2h/5.
\]
It gives rise to an elliptic fibration on $Y$.
Here $a_2$ serves as a bisection and $E$ (or $-E$) gives a half-pencil
since $a_2.E=1$.
Perpendicular to $E$, we find the sublattices
\[
A_1=\mathbb{Z} a_1 \;\;\; \text{ and } \;\;\; A_7 = \langle a_3,\hdots,a_9\rangle
\]
which thus comprise fiber components.
Since $A_9$ is has no integral overlattices,
$R_0=A_1+A_7$ is nodally primitive by Corollary \ref{cor:fibers},
and the reducible fibers have Kodaira types $I_8$ and $I_2$ or $III$.
It follows from \cite{MP} that
 $\mathop{\rm Jac}(Y)=X_{8211}$
 (with $I_2$, not $III$).
 Since the smooth rational bisection $a_2$ splits on the K3 cover,
 the claim of Proposition \ref{prop:base} follows from Lemma \ref{lem:bct}. \hfill \ensuremath{\Box}

 \subsection{Proof of Proposition \ref{prop:base} for $\boldsymbol{R=A_8+A_1}$}
 \label{ss:A_8+A_1}

 Let $Y$ be an Enriques surface of root type $R=A_8+A_1$.
 As indicated in Example \ref{ex:2-2},
 we have to distinguish 3 cases,
 depending on the primitive closure $R'$ of $R$ in $\mathop{\rm Num}(Y)$
 and the question whether $R'$, if it is a proper overlattice of $R$, is supported
on $(-2)$-curves on $Y$.

 \subsubsection{$R'=R$}
 \label{sss:A_8+A_1}

  If $R$ is primitive in $\mathop{\rm Num}(Y)$,
  then it is part of a index $18$ sublattice
  \[
  R + \mathbb{Z} h \subset \mathop{\rm Num}(Y),\;\;\; h^2=18.
  \]
 Consider the isotropic vector $E=a_1^\vee+2h/9$
 where $a_1^\vee \in A_8^\vee$.
 Then $E^2=0$,
 and since $a_1.E=1$,
 we obtain half-pencil and smooth rational bisection.
 Perpendicular to $E$, we find the sublattice $A_7+A_1$,
 nodally primitive by Corollary \ref{cor:fibers}
 so again $\mathop{\rm Jac}(Y)=X_{8211}$,
 and the splitting bisection $a_1$ finishes the job. \hfill \ensuremath{\Box}

\subsubsection{$R'=E_8+A_1$ supported on $(-2)$-curves}
\label{sss:E_8+A_1}

The gluing of $A_1$ and $\mathbb{Z} h, h^2=2$ is immediate.
The glue vector gives an elliptic fibration with singular fiber of type $II^*$
equipped with a bisection (given by the generator $a_1$ of $A_1$)
which only meets the simple component $e_9$ of the $II^*$ fiber.
By \cite{MP},
this leads to the extremal rational elliptic surfaces with further singular fibers either twice $I_1$
or once $II$,
and we conclude as before.
\hfill \ensuremath{\Box}

\subsubsection{$R'=E_8+A_1$ not supported on $(-2)$-curves}
\label{sss:E_8+A_1'}

In this case, the gluing vector $E$ is as above,
but perpendicular to $E$ there is only the root lattice $A_8$ supported on $(-2)$-curves.
By Corollary \ref{cor:fibers}, since $R$ is nodally primitive by assumption,
the only reducible fiber of the given elliptic fibration on $Y$ has type $I_9$.
This  uniquely determines the jacobian to be $X_{9111}$, see Table \ref{T0}.
In view of the splitting bisection $a_1$, Proposition \ref{prop:base} follows from Lemma \ref{lem:bct}.
\hfill \ensuremath{\Box}

\subsection{Proof of Proposition \ref{prop:base} for $R=A_7+2A_1$}

This is the first root type from Theorem \ref{thm:31}
which cannot embed primitively into $U+E_8$
because the length of the discriminant group $A_R$ exceeds one.
There are gluings
\[
A_7\subset E_7,\;\;\; A_7 + A_1 \subset D_8,
\]
for but length reasons they ultimately have to lead to
\[
R' = E_8 + A_1.
\]
Hence, we glue the summand $A_1$ which remains unaffected in the primitive
closure of $R$, to the vector $h$ of square $h^2=2$ as in \ref{sss:E_8+A_1}, \ref{sss:E_8+A_1'}.
For the reducible singular fibers of the elliptic fibration arising from $E$,
we have to distinguish which root lattices inside $R'$
are supported on $(-2)$-curves on $Y$.
Accordingly, we get $I_8 + I_2, III^* + I_2/III, I_4^*$ or $II^*$ by Corollary \ref{cor:fibers},
leading to the extremal rational elliptic surfaces $X_{8211}, X_{321}$ etc.
\hfill \ensuremath{\Box}

\begin{Remark}
Of course, the gluing is by no means unique.
In fact, for both types $A_8+A_1, A_7+2A_1$,
we will use a different isotropic vector in the next section
in order to ease computations
in working out the families of Enriques surfaces supporting them, see \ref{ss2:A_8+A_1}
and \ref{sss:A_7+2A_1}.
\end{Remark}

\subsection{Proof of Proposition \ref{prop:base} for all other root types}
\label{ss:glue}

For all these and the remaining root types $R$ on some Enriques surface $Y$ from Theorem \ref{thm:31}
the following table collects the  primitive closure
\[
R'\subset U+E_8 = \mathop{\rm Num}(Y),
\]
the isotropic vector $E$, given in terms of elements from the duals of the summands of $R$
and a fraction of the vector $h$ of square $h^2=-\mathop{\rm disc} R'$,
and the  root lattice
\[
R_0=R\cap E^\perp.
\]
Together these information suffice to read off the possible extremal elliptic surfaces
serving as $\mathop{\rm Jac}(Y)$ for the elliptic fibration arising from $E$.
Recall that the root type $R$ may also be supported on some integral overlattice,
as we shall exploit in Section \ref{s:number}.

\begin{table}
$$
\begin{array}{|cccc|}
\hline
R & R' & E & R_0\\
\hline
A_9 & A_9 & a_2^\vee + 2h/5 & A_7+A_1\\
A_8+A_1 & A_8+A_1 & (a_1^\vee,0)+2h/9 & A_7+A_1\\
& E_8+A_1 & (0,a_1^\vee) + h/2 & A_8\\
A_7 + A_2 & E_7 + A_2 & (a_2^\vee,0) + h/2 & A_5+A_2+A_1\\
A_7 + 2A_1 & E_8 + A_1 & (a_4^\vee,0,0) + h & 2A_3+2A_1\\
A_6+A_2+A_1 & A_6+A_2+A_1 & (a_1^\vee,0,0)+h/7 & A_5+A_2+A_1\\
A_5 + A_4 & A_5+A_4 & (0,a_2^\vee)+h/5 & A_5 + A_2 + A_1\\
A_5+A_3+A_1 & E_6+A_3 & (a_2^\vee,0,0)+h/3 & 2A_3+2A_1\\
A_5+2A_2 & E_7+A_2 & (0,0,a_1^\vee)+ h/3 & A_5+A_2+A_1\\
A_5+A_2+2A_1 & E_8+A_1 &(0,0,0,a_1^\vee)+h/2 & A_5+A_2+A_1\\
2A_4+A_1 & E_8 + A_1 & (0,0,a_1^\vee) + h/2 & 2A_4\\
A_4+A_3+2A_1 & A_4+D_5 & (a_1^\vee,0,0,0)+h/5 & 2A_3+2A_1\\
3A_3 & D_9 & (a_2^\vee,0,0) + h/2 & 2A_3+2A_1\\
2A_3+A_2+A_1 & E_7 + A_2 & (0,0,a_1^\vee,0) + h/3 & 2A_3+2A_1\\
2A_3+3A_1 & E_8+A_1 & (0,0,0,0,a_1^\vee)+h/2 & 2A_3+2A_1\\
A_3 + 3A_2 & A_3 + E_6 & (a_1^\vee,0,0,0) + h/4 & 4A_2 \\
D_9 & D_9 & d_9^\vee + 3h/4 & A_8\\
D_8+A_1 & E_8+A_1 & d_8^\vee + h & A_7+A_1\\
D_7+2A_1 & D_9 & (d_1^\vee,0,0)+h/2 & D_6+2A_1\\
D_6+A_3 & D_9 & (0,a_2^\vee)+h/2 & D_6+2A_1\\
D_6+A_2+A_1 & E_7 + A_2 & (0,a_1^\vee,0) + h/3 & D_6+2A_1\\
D_6+3A_1 & E_8+A_1 & (0,0,0,a_1^\vee) + h/2 & D_6+2A_1\\
D_5+A_4 & D_5+A_4 & (d_5^\vee,0) + h/4 & 2A_4\\
D_5+A_3+A_1 & E_8+A_1 & (0,0,a_1^\vee) + h/2 & D_5 + A_3\\
D_5+D_4 & D_9 & (0,d_1^\vee)+h/2 & D_5+A_3 \\
D_4+A_3+2A_1 & D_9 & (d_1^\vee,0,0,0)+h/2 & 2A_3+2A_1\\
2D_4+A_1 & E_8+A_1 & (0,0,a_1^\vee)+h/2 & 2D_4\\
E_8+A_1 & E_8+A_1 & (0,a_1^\vee) + h/2 & E_8\\
E_7+A_2 & E_7+A_2 & (e_7^\vee,0) + h/2 & E_6+A_2\\
E_7 + 2A_1 & E_8 + A_1 & (0,0,a_1^\vee) + h/2 & E_7 + A_1 \\
E_6+A_3 & E_6+A_3 & (e_6^\vee,0)+h/3 & D_5+A_3\\
E_6+A_2+A_1 & E_8+A_1 & (0,0,a_1^\vee)+h/2 & E_6+A_2\\
\hline
\end{array}
$$
\caption{Isotropic vectors and root lattices for the proof of Proposition \ref{prop:base}}
\label{T1}
\end{table}

To complete the proof of Proposition \ref{prop:base},
it suffices to argue with the $(-2)$ curve $C\subset Y$
whose dual vector is used for the gluing.
By construction, we have $C.E=1$ in either case,
so $E$ gives a half-pencil and $C$ gives a bisection for the fibration by Proposition \ref{prop:E}.
Hence Lemma \ref{lem:bct} implies that $Y$ arises from an Enriques involution of base change type
with base surface $\mathop{\rm Jac}(Y)$.

 \subsection{Relevant extremal rational elliptic surfaces}

 In this paragraph,
 we collect equations for all extremal rational elliptic surfaces needed for our purposes.
 The extended Weierstrass forms differ from other sources such as \cite{MP}
 as they are meant to make visible
 the singular fibers as much as possible (often obtained from Tate's (or Kubert's) normal form for elliptic curves
 with a rational point of given order, compare also Beauville's representations \cite{Beauville}).
 In particular, this will ease the process of
 endowing the quadratic twists with sections
 meeting the reducible fibers in a prescribed way
 (following Tate's algorithm \cite{Tate}).

 \begin{table}[ht!]
 $$
 \begin{array}{|cccc|}
 \hline
 \text{notation} & \text{Weierstrass eqn. } y^2 = f(x) & \text{sing. fibers} & \mathop{\rm MW}\\
 \hline
 X_{9111} & x^3+(tx+1)^2/4
 & I_9/\infty, I_1/t^3-27
 & \mathbb{Z}/3\mathbb{Z}\\
 X_{8211} & x(x^2+(t^2+2)x+1) & I_8/\infty, I_2/0, I_1/t^2+4 & \mathbb{Z}/4\mathbb{Z}\\
 X_{6321} & x(x^2+(t^2/4+t-2)x+1-t) &
 I_6/\infty, I_3/0, I_2/1, I_1/-8 &
 \mathbb{Z}/6\mathbb{Z}\\
 X_{5511} & x^3-tx^2+((1-t)x-t)^2/4 & I_5/0,\infty, I_1/t^2-11t-1 & \mathbb{Z}/5\mathbb{Z}\\
 X_{4422} & x(x-1)(x-t^2) & I_4/0,\infty, I_2/\pm 1 & \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\\
 X_{3333} &
 x^3 + (3tx+t^3-1)^2/4
 & I_3/\infty, t^3-1 & (\mathbb{Z}/3\mathbb{Z})^2\\

  X_{321} & x(x^2+x+t) & III^*/\infty, I_2/0, I_1/\frac14 & \mathbb{Z}/2\mathbb{Z}\\
 X_{222} & x(x-1)(x-t) & I_2^*/\infty, I_2/0,1 & (\mathbb{Z}/2\mathbb{Z})^2\\
 X_{211} & x^3+x^2+t & II^*/\infty, I_1/0, -\frac 4{27} & 0\\
 X_{141} & x(x^2+t(t+2)x+t^2) & I_4/\infty, I_1^*/0, I_1/-4 & \mathbb{Z}/4\mathbb{Z}\\
 X_{\lambda} & x(x^2+tx+\lambda t^2 x) & I_0^*/0,\infty & (\mathbb{Z}/2\mathbb{Z})^2\\
 \hline
 \end{array}
 $$
 \caption{Extremal rational elliptic surfaces}
 \label{T0}
 \end{table}

 Note that the
 $2$-torsion sections are always located at zeros of $f(x)$;
 here we made sure to always normalise one of them, if any over $K(t)$,  to $(0,0)$.

 \begin{Remark}
 \label{rem}
 The attentive reader will notice that the isotrivial extremal rational elliptic surfaces
 $X_{22}, X_{33}, X_{44}$ (in the notation of \cite{MP})
 do not appear in the above table,
 although they are not distinguished from $X_{211}, X_{321}, X_{431}$
 by the root lattices associated to the singular fibers
 (see the two cases often appearing in the preceeding paragraphs).
 The reason for their absence
 lies in the fact that the quadratic base changes of the latter have indeed two moduli,
 while the former only yield one parameter (since there is still a scaling in the base variable),
 and in fact they form a subfamily of the latter (cf. the discussion for $X_{431}$ and $X_{44}$ in \cite[\S 3.3]{RS}).
 \end{Remark}

 \section{One-dimensional families}
 \label{s:fam}

 This section completes the proof of Theorem \ref{thm},
 except for \textit{(ii)}.
 The overall strategy is clear:
 given an Enriques surface $Y$ supporting a root lattice from Theorem \ref{thm:31},
 Proposition \ref{prop:base} tells us that $Y$ arises from an extremal rational elliptic surface $S$
 through the base change type construction of Enriques involutions.
 Indeed, Table \ref{T1} not only predicts  $S$ (although sometimes there are a few possible choices to consider,
 depending on the overlattices supported on $(-2)$-curves),
 but also pinpoints the bisection on $Y$
 and how it intersects the reducible singular fibers in the nodally primitive case by Corollary \ref{cor:fibers}.
 This information easily translates into the section $P$ on the K3 cover $X$,
 and the underlying section $P'$ on some quadratic twist $S'$ of $S$.
 Recall from Section \ref{s:tech} how the shape of $P'$ is predicted:
 as a polynomial $x(P')$ of degree $2$ (see \eqref{eq:x(P')}).
 This puts us in the comfortable position
 to aim directly for a parametrization of all quadratic twists $S'$ admitting a section $P'$ of the required shape.
 The results will be displayed in Table \ref{T2}.

 Before covering all root types,
 we start out by a detailed continuation of the examples
 discussed in the previous section,
 that is for root types $A_9, A_8+A_1$ and $E_8+A_1$.
 This should  give  a clear idea how to carry out the computations.
 For most other types,
 we omit the details
 (which are available from the author upon request),
 and only list the resulting families:
 Table \ref{T2} phrases them completely in terms of the quadratic twist $S'$, see \ref{ss:T3}.
 We then justify Table \ref{T2} by supplying pointers to the few critical steps
 in the computations.
In \ref{ss:proof}, we collect all the additional data needed to prove Theorem \ref{thm}.

 \subsection{Proof for root type $R=A_9$}
 \label{ss2:A_9}

 We continue the argument from \ref{ss:A_9}, retaining the notation.
 On the K3 cover $X$, the pullback $\pi^* a_2$ splits into sections $O, P$
which generically meet adjacent components in each reducible fiber.
More precisely, unless one of the reducible fibers of $Y$ is ramified,
$a_2$ meets the fiber components $a_3$ and the omitted one in the extended Dynkin diagram $\tilde A_7$
corresponding to $I_8$,
and likewise with $a_1$ in $\tilde A_1$.
Thus we exactly have the picture from Figure \ref{fig:A_9}.

In view of Theorem \ref{thm},
we claim that the Enriques surfaces of root type $A_9$ come exactly
in the one-dimensional family from Example \ref{ex:1-2}.
In order to see this,
it remains to study the  cases with multiple reducible fibers.
Note that  a ramified fiber of type $I_n$ on $Y$
 is replaced by a single fiber of type $I_{2n}$ on $X$;
 necessarily it is met by $P$ at the component opposite to the identity component
 since $P$ always intersects the ramified fibers in 2-torsion points
 (compare the logarithmic transformation!),
 but translation by $P$ does not fix any fiber component as a set.
 We continue to argue with the height from the theory of Mordell-Weil lattices \cite{ShMW},
 applied to $X$,
 which generically returns
 \[
 h(P) = 4 - 2\cdot \frac 78 -2 \cdot \frac 12 = \frac 54 .
 \]
 If the $I_8$ fiber were ramified,
 then the height would degenerate to
 \begin{eqnarray}
 \label{eq:ht}
 h(P) = 4-\frac{8\cdot 8}{16} - (\text{further non-zero correction terms}) <0,
 \end{eqnarray}
 which is impossible.
 Otherwise, if the $I_2$ fiber is ramified, then the height stays the same,
 but the Picard number of $X$ goes up to $20$ due to the fiber degeneration of two $I_2$'s
 merging to one $I_4$ fiber.
 In the family from Example \ref{ex:1-2},
 this visibly occurs at $\mu=\pm 1$
 which confirms the claim.
 \hfill \ensuremath{\Box}

 \begin{Remark}
 Alternatively, one could consider the isotropic vector $a_1^\vee+3h/10$
 which gives an extremal fibration with an $I_9$ fiber on $Y$,
 or $a_5^\vee+h/2$
 which gives an extremal fibration with two $I_5$ fibers on $Y$.
 In either case, the corresponding explicit calculations
 can be carried out without  much difficulty,
 but the resulting equations turn out more complicated than in Example \ref{ex:1-2}.
 \end{Remark}

\subsection{Proof for root type $R=A_8+A_1$}
\label{ss2:A_8+A_1}
\label{ss2:E_8+A_1}
\label{ss:not}

In \ref{ss:A_8+A_1},
we distinguished three cases:
\begin{enumerate}
\item
\label{it1}
$R$ primitive in $\mathop{\rm Num}(Y)$,
\item
\label{it2}
$R'=E_8+A_1$ supported on $(-2)$-curves on $Y$,
\item\label{it3}
$R'=E_8+A_1$ not supported on $(-2)$-curves, i.e.~$R$ nodally primitive.
\end{enumerate}
For each case, starting from the isotropic vector $E$ from Table \ref{T1},
one can work out the Enriques involution of base change type,
starting from the extremal rational elliptic surface $X_{8211}, X_{211},$ resp. $X_{9111}$.
In particular, this shows that each case occurs in a one-dimensional family.
However, there is a surprising symmetry which the above approach does not detect:

\begin{Lemma}
\label{lem:iso}
An Enriques surface supports the primitive root type $A_8+A_1$ (case \eqref{it1})
if and only if it supports the imprimitive, but nodally primitive root type $A_8+A_1$
(case \eqref{it3}).
\end{Lemma}

Given an Enriques surface $Y$ supporting the root type $A_8+A_1$,
we argue with the dual vector $a_3^\vee\in A_8^\vee$ of square $(a_3^\vee)^2=-2$.
Note that as an abstract lattice $E_8=\langle A_8,a_3^\vee\rangle$.
Consider the isotropic vector
\[
E = \begin{cases}
a_3^\vee + h/3, & \text { in case \eqref{it1}},\\
a_3^\vee + h, & \text{ in cases \eqref{it2}, \eqref{it3}}
\end{cases}
\]
(after checking that in case \eqref{it1}, we indeed have $E\in\mathop{\rm Num}(Y)$).
Since $a_3.E=1$, $E$ gives the half-pencil of some elliptic fibration on $Y$ and $a_3$ a bisection
by Proposition \ref{prop:E}.
Perpendicular to $E$ in $R$, we find $A_1$ and inside $A_8$
\[
A_2 = \langle a_1,a_2\rangle,\;\;\; A_5 = \langle a_4,\hdots,a_8\rangle.
\]
By Criterion \ref{crit}, the rational elliptic surface $\mathop{\rm Jac}(Y)$ is extremal.
We distinguish the possible cases.

\subsubsection{$\mathop{\rm Jac}(Y) = X_{6321}$}
\label{sss:6321}

If $R$ is nodally primitive,
i.e. in cases \eqref{it1}, \eqref{it3},
the fibration has singular fibers of Kodaira types $I_6, I_3, I_2$
by Corollary \ref{cor:fibers},
so $\mathop{\rm Jac}(Y) = X_{6321}$ by \cite{MP}.
By Lemma \ref{lem:bi}, the bisection $a_3$ meets both $I_6$ and $I_3$ fibers in two adjacent components
(unless they are multiple).
On the quadratic twist $S'$ of $X_{6321}$, this uniquely leads to a section $P'$ with
\[
x(P') = \mu t+1
\]
which determines $q=(\mu t+1)(\mu t + (2\mu+1)^2)$.
The degenerate case with double $I_3$ fiber occurs at $\mu=-1/2$
while a double $I_6$ fiber cannot occur by inspection of the height pairing similar to \eqref{eq:ht}.

\subsubsection{Proof of Lemma \ref{lem:iso}}
In conclusion of \ref{sss:6321}, both isotropic vectors for cases \eqref{it1}, \eqref{it3} lead to the same family of elliptic fibrations,
jacobians, quadratic twists, etc.
Lemma \ref{lem:iso} follows.
\hfill \ensuremath{\Box}

\subsubsection{Explicit equivalence of \eqref{it1} and \eqref{it3}}

As a sanity check,
we explain how to show the equivalence of the three fibrations arising from $X_{9111}, X_{8211}$ and $X_{6321}$
in a purely geometric way.
To this end, it suffices to trace back $(-2)$ curves
as we sketch for one case.

Start from \eqref{it3}.
According to Table \ref{T1},
there is an elliptic fibration on $Y$
with singular fiber of type $I_9$
and smooth rational bisection $C$ meeting the fiber at a single component.
Translation by a $3$-torsion section on the K3-cover $X$ induces an automorphism $\sigma$ of $Y$.
Using the height pairing on $X$, one verifies that $C.\sigma(C)=2$.
That is, $D=C+\sigma(C)$ defines a divisor of Kodaira type $I_2$.
Perpendicular to this divisor, there are root lattices $A_2+A_5$ as depicted in the following figure.

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(80,70)(8,-5)

\multiput(60,0)(0,20){4}{\circle*{1.5}}
\put(60,0){\line(0,1){60}}

\multiput(0,30)(20,10){3}{\circle*{1.5}}
\put(0,30){\line(2,1){60}}

\multiput(0,30)(20,-10){3}{\circle*{1.5}}
\put(0,30){\line(2,-1){60}}

\put(80,0){\circle*{1.5}}
\put(83,-2){$\sigma(C)$}

\put(80,60){\circle*{1.5}}
\put(83,58){$C$}

\put(79.5,0){\line(0,1){60}}
\put(80.5,0){\line(0,1){60}}

\put(60,-.5){\line(1,0){20}}
\put(60,.5){\line(1,0){20}}
\put(60,59.5){\line(1,0){20}}
\put(60,60.5){\line(1,0){20}}

\end{picture}
\end{figure}

For the elliptic fibration induced by $|D|$ (or a priori $|2D|$), this implies that $\mathop{\rm Jac}(Y) = X_{6321}$ as desired.
It remains to prove that the fiber components $\Theta_0,\Theta_3$ give bisections
to conclude using Lemma \ref{lem:bct} that the fibration is exactly the one from \ref{sss:6321}.
Otherwise the $I_2$ fiber would be ramified and $\Theta_0,\Theta_3$ would be four-sections.
But then the Gram matrix with entries for the 11 fiber components and the 2 four-sections would have rank $12$,
contradiction.

The step from $X_{6321}$ to $X_{8211}$ is similar,
again using $3$-torsion sections on the K3 cover $X$.
We leave the details to the reader.

\subsubsection{$\mathop{\rm Jac}(Y)=X_{321}$}
\label{sss:321}

For completeness, we continue to discuss the case \eqref{it2}
where the primitive closure $R'$ of $R$ is in fact supported on $(-2)$-curves on $Y$,
i.e.~the dual vector $a_3^\vee\in A_8^\vee$ corresponds to a smooth rational curve on $Y$.
In terms of the basis of $E_8$ from \ref{ss:E_l},
this translates as
\[
e_1=a_3^\vee, \;\; e_i=a_{i-1}  \; (i=2,\hdots,8).
\]
Note that $e_1^\perp \cong E_7 \subset E_8$,
i.e.~$R_0=A_2+A_5$ is not nodally primitive.
By Corollary \ref{cor:fibers}, the elliptic fibration has singular fibers of types $III^*$ and $I_2$ or $III$.
By Proposition \ref{prop:E},
the $(-2)$-curve $e_4$ gives a bisection of the elliptic fibration on $Y$, and $E$ a half-pencil.
We emphasize, however, that neither $E$ nor $-E$, no matter which is effective, is base-point free,
since either $e_1$ or $e_4$ meets it negatively.
After subtracting the base locus, however, one indeed arrives at the perpendicular lattice $E_7$ with smooth rational curve,
aka bisection, attached to $e_7$ (and perpendicular to the orthogonal summand $A_1$).
By Lemma \ref{lem:bi},
this translates into the following diagram of smooth rational curves:

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(120,40)(0,0)
\multiput(20,0)(20,0){3}{\circle*{1.5}}
\put(20,0){\line(1,0){40}}

\multiput(20,40)(20,0){3}{\circle*{1.5}}
\put(20,40){\line(1,0){40}}

\put(20,0){\line(0,1){40}}
\put(20,20){\line(-1,0){20}}

\put(0,20){\circle*{1.5}}
\put(20,20){\circle*{1.5}}

\put(60,40){\line(1,-1){20}}
\put(60,0){\line(1,1){20}}

\put(80,20){\line(1,0){20}}

\put(100,19){\line(1,0){20}}
\put(100,21){\line(1,0){20}}

\multiput(80,20)(20,0){3}{\circle*{1.5}}
\put(121,14){$A_1$}

\end{picture}
\caption{$E_8+A_1$ supported on fibers of type $III^*, I_2$ and bisection}
\label{fig:321}
\end{figure}

There is one point in the above figure that might require clarification.
Namely, a priori one might expect that the bisection $a_3$
meets the $I_2$ fiber to the right (containing the summand $A_1$ perpendicular to $a_3$)
doubly in a single component.
To rule this out, note that in view of the $III^*$ fiber, we have $\mathop{\rm Jac}(Y)=X_{321}$  by \cite{MP}
(or $X_{33}$ which we shall rule out below, compare Remark \ref{rem}).
Thus we can solve for a quadratic twist of $X_{321}$
to admit an integral section $P'$ inducing $a_3$ as in Section \ref{s:tech}.
For the extended Weierstrass form from Table \ref{T0},
the condition for $P'$ to meet the non-trivial simple component of the $III^*$ fiber
of the quadratic twist translates as triple vanishing of $x(P')$ at $\infty$.
But since $x(P')$ has degree two by \eqref{eq:x(P')}, this gives $x(P')\equiv 0$,
i.e. $P'$ is the 2-torsion section $(0,0)$.
In particular, $P'$ meets the $I_2$ fiber at the non-identity component;
the same holds for the bisection $a_3$ on $Y$
induced by the Enriques involution of base change type
unless the $I_2$ fiber is ramified as depicted in Figure \ref{fig:321}.

Note that the same argument goes through if $\mathop{\rm Jac}(Y)$ were to be $X_{33}$,
but then we would arrive at a multiple $III$ fiber which is impossible as stated above.

\begin{Remark}
The family supporting the root type $E_8+A_1$ is remarkable since it comprises Enriques surfaces with finite automorphism group
(Kondo's type I in \cite{Kondo-Enriques}).
We remind the reader  that an Enriques surface has finite automorphism group
if and only if the set of smooth rational curves is non-empty, yet finite.
\end{Remark}

\begin{Corollary}
\label{cor:A8+A1}
The root type $A_8+A_1$ is supported on two distinct families
of Enriques surfaces.
\end{Corollary}

\begin{proof}
The above discussion shows that there are at most two components.
It remains to prove that
 \eqref{it1} and \eqref{it2} are indeed not equivalent.
 For instance, this can be read off from the  transcendental lattices of the covering K3 surfaces,
 or from their discriminants.
The former turn out to be generically
 $U+\langle 36\rangle$ for \eqref{it1} and \eqref{it3},
 and  $U+\langle 4\rangle$ for \eqref{it2}
 (calculated using discriminant forms following Nikulin \cite{Nikulin})
\end{proof}

\subsection{Overview of the parametrisations}

Roughly, there are three cases when going for the explicit parametrisations,
starting from an extremal rational elliptic surface $S$.
Recall that we want to endow a quadratic twist $S'$ of $S$
with an integral section $P'$ of prescribed shape (drawing on Lemma \ref{lem:bi}).

\subsubsection{}
If $P'$ meets two different singular fibers (other than the two $I_0^*$ fibers originating from the twist) non-trivially,
 or one fiber at a fiber component not adjacent to the identity component,
 then this determines $x(P')$ up to one parameter
 and directly gives the desired family of Enriques surfaces
 as seen in Example \ref{ex:1-2}.

 \subsubsection{}
 \label{sss:1f}
 If $P'$ meets  one  singular fibers (other than the two $I_0^*$ fibers originating from the twist)
 in a component adjacent to the identity component,
then this gives one condition on $x(P')$ and thus leaves two parameters.
Substituting into the RHS of the extended Weierstrass form of $S$ from Table \ref{T0}
gives a square factor and  a degree $4$ polynomial $g$ in $t$.
Then we simply solve for the discriminant of $g$ to vanish
while preserving the required shape of $P'$
and ensuring that $q$ does not degenerate.

\subsubsection{}
\label{sss:0f}
If $P'$ meets no singular fibers other than the two $I_0^*$ fibers originating from the twist
non-trivially,
then this gives no new conditions on $x(P')$
and there are three parameters to master.
In general, this complicates computations substantially,
but one can still handle them.

\subsubsection{Effect of two-torsion}
\label{sss:2t}

Even in the last case,
the situation changes drastically in the presence of two-torsion
in the Mordell-Weil group.
Normalising the twisted Weierstrass form of $S'$ as in Table \ref{T0} to
\[
S':\;\;\; qy^2 = x(x^2+a_2x+a_4),\;\;\; a_i\in K[t],\;\; \deg a_i\leq i,
\]
we find that $x(P')$ and $a_4$ are coprime,
since otherwise $P'$ would intersect the corresponding singular fiber non-trivially.
In order to be able to solve for $y(P')$,
we dedude that
\begin{itemize}
\item
either $x(P')$ is a perfect square
(thus reducing the number of parameters to two,
so that one can solve as in \ref{sss:1f})
\item
or
$x(P')=\alpha q$ for a scalar $\alpha\in K$.
Here one can solve directly for the remaining factor on the RHS upon substitution,
 $x(P')^2+a_2 x(P')+a_4$
to give a perfect square.
\end{itemize}

\subsection{Explicit equations}
\label{ss:T3}

 The following table collects all Enriques surfaces with root type from Theorem \ref{thm:31}.
 As it happens, they  all appear in one-dimensional families as stated in Theorem \ref{thm}.
 The base is always $\mathbb{P}^1$,
 with varying parameter $\mu, \lambda, s$ or $u$.
The Enriques surfaces $Y$ are given in terms of
\begin{itemize}
\item
an extremal rational elliptic surface $S$ which is
the jacobian $\mathop{\rm Jac}(Y)$ of an elliptic fibration on $Y$
by Proposition \ref{prop:base},
\item
its quadratic twist $S'$, given by a quadratic polynomial $q\in K(\mathbb{P}^1)[t]$,
\item
a section $P'$ of $S'$ (given by $x(P')$)
which induces the Enriques involution of base change type on the K3 cover $X$ of $Y$.
\end{itemize}
 The reader should be advised that there always is finite set of places of $\mathbb{P}^1$
 where the parameter ceases to give an Enriques surface due to degenerations.
 We do not list these places explicitly, as they are easily calculated.

 \begin{table}[ht!]
 $$
 \begin{array}{|lccc|}
 \hline
 \text{root type} & {\mathop{\rm Jac}(Y)} & \text{section:} \, x(P') & \text{quadr.~twist:} \, q\\
 \hline
A_9 & X_{8211} & \mu t -1 & (\mu t-1) (\mu t +\mu^2-1)\\
A_8 + A_1 & X_{6321} & \mu t + 1 & (\mu t+1)(\mu t + (2\mu+1)^2)
\\
A_7+A_2 & X_{6321} & \mu(t-1)
&
\mu t^2+4\mu (\mu+1) t-4(\mu+1)^2
\\
A_7+2A_1 & X_{4422} & \mu t & (\mu t-1)(t-\mu)\\
A_6+A_2+A_1 & X_{6321} &
-(ut-4u^2+3u-1)/(u^2(u+1))
&\\
&&\multicolumn{2}{r|}{((u^2+u)t-(u-1)^2)(ut-4u^2+3u-1)}\\
A_5+A_4 & X_{6321} & (t-1)(\mu t-1) & (\mu t-1)(\mu(4\mu+1)t-(2\mu+1)^2)\\
A_5+A_3+A_1 & X_{4422} &
\mu t+1-\mu &
(\mu t+1-\mu)(t+1-\mu)
\\
A_5+2A_2 & X_{6321} & 27(t+u)(t-1)/((u-8)(u+1)^2) &
(t+u)(3(u+4)t-(u-2)^2)\\
A_5+A_2+2A_1 & X_{6321} & 27(t-u)^2/((u-1)(u+8)^2) &\\
&&\multicolumn{2}{r|}{
3(u+11)t^2-4(u^2+13u+4)t+4(u-4)^2}
\\
2A_4+A_1 & X_{5511} &
\multicolumn{2}{l|}{-(s^5t^2-s^3(11s^2-15s+5)t-(s-1)^5)/(5s^2-5s+1)^2}
\\
&&\multicolumn{2}{r|}
{s^2(4s-1)t^2-2(2s-1)(11s^2-11s+2)t-(4s-3)(s-1)^2}
\\
A_4+A_3+2A_1 & X_{4422} &
4\mu(t+\mu) &
(t+\mu)
(4\mu t-4\mu^2-1)\\
3A_3 & X_{4422} & \mu t^2-\mu+1
& \mu t^2-\mu+1 \\
2A_3 + A_2 + A_1 & X_{4422} & (\mu t-\mu+1)^2 &
(\mu t-\mu+2)(\mu t+t-\mu+1)
\\
2A_3+3A_1 & X_{4422} &
-(t^2+4\mu t+1)/(4\mu^2-1)&
t^2+4\mu t+1
\\
A_3+3A_2
& X_{3333} & \multicolumn{2}{l|}{(t-1)((s^3+6s^2+9s+3)t+s^3+3s^2-3)/s^2(2s+3)}\\
&&\multicolumn{2}{r|}{((6s+3)(s+1)^2t^2+(8s^3+12s^2-6s-6)t+4s^3-6s+3)}\\
D_9 & X_{9111} & \mu &
\mu^2t^2+2\mu t+4\mu^3+1
\\
D_8+A_1 & X_{8211} & \mu & \mu t^2 + (\mu+1)^2
\\
 D_7+2A_1 & X_{222} & t+\mu & (t+\mu)(t+\mu-1)
 \\
 D_6+A_3 & X_{222} & (\mu t+1-\mu)t & (\mu t+1-\mu)(\mu t+1)\\
 D_6+A_2+A_1 & X_{222} & -4\lambda t(\lambda t-1)
 & (\lambda t-1)(4\lambda^2 t-4\lambda+1)
 \\
 D_6+3A_1 & X_{222} & 0 & t(t-\lambda)\\
 D_5+A_4 & X_{5511} & \multicolumn{2}{r|}{ \mu \;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;
 (\mu+1)^2t^2-2\mu(3\mu+1)t+\mu^2(4\mu+1)}
 \\
 D_5 + A_3 + A_1

 & X_{4422} & 1 & (t-1)(t-\lambda)
 \\
 D_5+D_4 & X_{141} & \mu & (\mu+1)t^2+2\mu t + \mu^2\\
 D_4+A_3+2A_1 & X_{4422} &\mu & t^2-\mu\\
  2D_4+A_1 & \multicolumn{2}{l}{\;X_{-4\mu(4\mu+1)} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (t+\mu)^2} & t^2+(6\mu+1)t+\mu^2
 \\
 E_8+A_1 & X_{321} & 0 & t(t-\lambda)\\
 E_7+A_2 & X_{6321} & 0 & t-\lambda\\
 E_7+2A_1 & X_{222} & 0 & t(t-\lambda)\\
 E_6+A_3 & X_{141} & (\mu t-1)t & (\mu t-1)(\mu(\mu+1) t-1)
 \\
 E_6+A_2+A_1 & X_{6321}  & 0 & (t-1)(t-\lambda)
 \\
 \hline
 \end{array}
 $$
 \caption{Explicit families of Enriques surfaces supporting the 31 maximal root types}
 \label{T2}
 \end{table}

 \subsection{Proof of Theorem \ref{thm} \textit{(i), (iii), (iv)}}
 \label{ss:proof}

Table \ref{T2} lists all families of Enriques surfaces
that the direct approach sketched in the preceding sections yields.
Throughout we assume that the root type $R_0$ is nodally primitive
so that Lemma \ref{prop:fibers} kicks in to determine the singular fibers from $R_0$,
and Lemma \ref{lem:bi} pinpoints the intersection pattern with the smooth rational bisection.
(Indeed, except for \ref{sss2}, it suffices to assume that $R$ is nodally primitive by Corollary \ref{cor:fibers},
compare Remark \ref{rem:prim}.)

There are eight notable exceptions where for different reasons Table \ref{T2} does not give the families
arising from Proposition \ref{prop:base} and Table \ref{T1},
starting with $A_8+A_1$ as discussed in \ref{ss:not}.
The reasons either lie in the presentation,
i.e. a different approach gives a significantly nicer parametrisation (\ref{sss3} - \ref{sss6}),
or in the problem of distinguishing possibly different components of the moduli space
-- this is done in \ref{sss:A_7+2A_1}, \ref{sss2} similarly to \ref{ss2:A_8+A_1}.
In the sequel we sketch the details.

\subsubsection{Root type $A_7+2A_1$}
\label{sss:A_7+2A_1}

Building on Corollary \ref{cor:fibers} and the assumption that $R$ is nodally primitive,
Proposition \ref{prop:base} and Table \ref{T1} suggest to work with
the base change technique starting from $X_{8211}$.
Due to the 2-torsion section, we can use \ref{sss:2t} to search for quadratic twists with an integral section
of height $4$.
As it happens, this leads to two distinct parametrisations.
In \ref{ss:A7+2A1}, we will deal with them directly;
here we instead apply our methods to a different isotropic vector, namely
$E=(a_4^\vee,0,0)+h$ with orthogonal root lattice $R_0=2A_3+2A_1$.
By Corollary \ref{cor:fibers}, we have $\mathop{\rm Jac}(Y)=X_{4422}$.
Based on Lemma \ref{lem:bi},
quadratic twist and section
are immediate as displayed in Table \ref{T2}.
For further ways, to realize this root type, corresponding to integral overlattices of $R$ supported
on $(-2)$-curves on $Y$, see Remark \ref{rem:prim} and especially \ref{ss:A7+2A1}.

\subsubsection{Root type $D_6+3A_1$}
\label{sss2}

Proposition \ref{prop:base} and Table \ref{T1} tell us that
there is an elliptic fibration on $Y$ with
root lattices $D_6+2A_1$ embedding into the singular fibers.
By Criterion \ref{crit},
the rational elliptic surface $\mathop{\rm Jac}(Y)$ is extremal.
Note that if $R_0$ is nodally primitive,
then as in Lemma \ref{lem:bi}
none of these $(-2)$-curves is met by the smooth rational bisection (given by a generator $a_1$ of the third summand $A_1$).
From \cite{MP} we can already deduce
that the reducible fibers
have types $I_2^*, I_2, I_2$,
i.e. $\mathop{\rm Jac}(Y) = X_{222}$.
On the quadratic twist $S'$,
this leads to an integral section $P'$ of height $h(P')=4$ as in \ref{sss:0f},
but with the advantage of having a 2-torsion section, so that \ref{sss:2t} kicks in.
Indeed one can easily parametrise the quadratic twists with section $P'$ as required,
but this leads to two parametrisations,
one for each case from \ref{sss:2t}:
\[
x(P') = (t+\lambda)^2/4\lambda
\;\;\;
\text{ and } \;\;\; x(P') = (\mu^2t^2-2\mu(\mu-1)t+(\mu+1)^2)/4\mu.
\]
To see that these parametrisations give the same family of Enriques surfaces
(without having to determine them explicitly, in fact),
we switch to an auxiliary fibration as follows.
Consider the divisor $D=d_0+a_1$ of Kodaira type $I_2$ on $Y$
where $d_0$ denotes the component of the original $I_2^*$ fiber met doubly by the bisection $a_1$.
Since $d_2.D=1$,
$D$ is a half-pencil and $d_2$ a bisection for the elliptic fibration
induced by $|2D|$.
Naturally this has $D$ as a multiple singular fiber and the other singular fibers containing
what's orthogonal to $D$:
\begin{eqnarray}
\label{eq:D4+3A1}
D_4=\langle d_3,\hdots, d_6\rangle,\;\; A_1=\mathbb{Z} d_1,\;\; 2A_1
\end{eqnarray}
where the last two summands stem from the original root type
(or equivalently from the two $I_2$ fibers -- the component not met by $a_1$).
By criterion \ref{crit},
$\mathop{\rm Jac}(Y)$ is an extremal rational elliptic surface,
so we can use \cite{MP} to work out that it can only be $X_{222}$ again.
In particular, there is a fiber of type $I_2^*$ comprising $D_4$, two $A_1$'s and a connecting $(-2)$-curve.
Recall that the bisection $d_2$ meets the fiber in simple components;
by inspection, these have to be $d_1, d_3$, since $d_2$ is perpendicular to all other $(-2)$-curves in \eqref{eq:D4+3A1}.
The next figure sketches the resulting configuration of smooth rational curves (up to exchanging $d_5, d_6$).

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(90,78)(0,0)

\multiput(0,0)(30,0){2}{\circle*{1.5}}
\put(0,0){\line(1,1){15}}
\put(30,0){\line(-1,1){15}}

\put(-5,4){$d_6$}
\put(32,-4){$d_3$}

\multiput(0,70)(30,0){2}{\circle*{1.5}}

\put(0,70){\line(1,-1){15}}
\put(30,70){\line(-1,-1){15}}

\put(-5,63){$A_1$}
\put(32,71){$d_1$}

\multiput(15,15)(0,20){3}{\circle*{1.5}}
\put(15,15){\line(0,1){40}}

\put(17,16){$d_4$}

\put(17,36){$d_5$}

\put(47.5,35){\circle*{1.5}}
\put(38,33){$d_2$}

\put(30,0){\line(1,2){17.5}}

\put(30,70){\line(1,-2){17.5}}

\put(47.5,35){\line(1,-1){15}}

\multiput(62.5,20)(15,-15){2}{\circle*{1.5}}
\put(64,22){$d_0$}
\put(79,7){$a_1$}

\put(62.1,19.6){\line(1,-1){15}}
\put(62.9,20.4){\line(1,-1){15}}

\multiput(62.5,50)(15,15){2}{\circle*{1.5}}
\put(79,61){$A_1$}

\put(47.1,35.4){\line(1,1){30}}
\put(47.9,34.6){\line(1,1){30}}

\end{picture}
\caption{Induced fibration for $D_6+3A_1$}
\label{fig:D6+3A1}
\end{figure}

On the K3  cover $X$, the bisection $d_2$ splits into disjoint sections $O,P$.
Since there are correction terms  twice $3/2$ from the two $I_2^*$ fibers and $1$ from the ramified $I_4$ fiber,
we infer that $P$ has height zero.
That is, $P$ is two-torsion, so up to exchanging the two $I_2$ fibers, $x(P')$ is as in
Table \ref{T2}.

\begin{Remark}
\label{rem:prim}
We point out that the $(-2)$-curves in Figure \ref{fig:D6+3A1}
also support the root types $E_7+2A_1$ and $A_7+2A_1$:
for $E_7$,
starting from $d_6$, go up including both branches with $A_1$ and $d_1$
and end at $d_2$ to find $E_7$;
for $A_7$ exchange the $(-2)$-curve  $d_5$ with $d_3$ in the above tree.
The two perpendicular $A_1$'s are always located at the far right vertices of the graph.
This is a rather special situation:
the root type $D_6+3A_1$ is not nodally primitive on any Enriques surface,
but automatically comes with the overlattice $E_7+2A_1$
supported on $(-2)$-curves
(as reflected in Table \ref{T2}).
\end{Remark}

\subsubsection{Root type $D_5+A_3+A_1$}
\label{sss3}

By Proposition \ref{prop:base} and Table \ref{T1},
any Enriques surface $Y$ with this root type originates from a quadratic twist of $X_{141}$
with singular fibers $I_1^*, I_4$ and integral section $P'$ of height $h(P')=4$
which can be determined explicitly as outlined in \ref{sss:0f}, \ref{sss:2t}.

For a nicer representative, we use the Kodaira divisor $D=d_0+a_1$ of type $I_2$ on $Y$
where $a_1$ denotes the generator of the orthogonal summand $A_1$,
and $d_0$ denotes the component of the $I_1^*$ fiber met doubly by $a_1$.
By Proposition \ref{prop:E},
$|2D|$ induces another elliptic fibration on $Y$,
with singular fibers of types $I_4, I_4, I_2$
given by the perpendicular root lattices $A_3$ from the original orthogonal sum and
$A_1=\mathbb{Z} d_1, A_3 = \langle d_3,\hdots,d_5\rangle\subset D_5$.
This shows that $\mathop{\rm Jac}(Y)=X_{4422}$.

In detail, $d_2$ serves as a bisection,
and since $d_2.D=1$, $D$ is in fact a half-pencil.
By inspection of the height pairing,
$d_2$ splits into zero and two-torsion section on the K3 cover $X$.
Up to symmetry, we can thus locate the ramified $I_2$ fiber at $t=1$
and assume that the two sections associated to $d_2$ are $O$ and $(1,0)$,
as displayed in Table \ref{T2}.

\subsubsection{Root type $E_8+A_1$}

This type has been discussed in \ref{ss2:E_8+A_1}
where also the entry from Table \ref{T2} was deduced.

\subsubsection{Root type $E_7+A_2$}

In this case, there is a unique parametrisation starting from $X_{321}$
as indicated in Table \ref{T1}, but it is comparably complicated
relative to the given representation.

\subsubsection{Root type $E_7+2A_1$}

Again there is a unique parametrisation involving $X_{321}$,
but the given one which was pointed out in Remark \ref{rem:prim},
is much simpler.

\subsubsection{Root type $E_6+A_2+A_1$}
\label{sss6}

Here also, the calculation of the parametrisation starting from $X_{431}$
becomes rather complicated,
but it leads to a single family.
The present representation was chosen for simplicity.

\subsubsection{Summary}
\label{ss:summary}

For each root type $R$ from Theorem \ref{thm:31},
we use  Proposition \ref{prop:base} and Table \ref{T1}
to parametrise the Enriques surfaces supporting $R$.
Table \ref{T2} lists exactly the families resulting from the assumption that $R_0$ is nodally primitive
so that the singular fibers are determined by Lemma \ref{prop:fibers}
(except for those families where we chose to list a more convenient model
as explained above).

A few times, allowing for  singular fibers with multiplicity two leads to additional families as in \ref{ss:not}.
By inspection, these always support a root lattice $R'$ which is an overlattice of $R$.
Hence these families  file under the root type $R'$.
This proves Theorem \ref{thm} \textit{(i), (iii), (iv)}.
\hfill \ensuremath{\Box}

We will give a brief account of the problem of the overlattices,
and thus of the number of components of the moduli spaces, in the next section.
This will complete the proof of Theorem \ref{thm} by addressing \textit{(ii)}.

\section{Components of the moduli spaces}
\label{s:number}

In order to complete the proof of Theorem \ref{thm},
it remains to study the number of components of the moduli spaces
for each of the root types from Theorem \ref{thm:31}.
Recall from Corollary \ref{cor:A8+A1}
that the root type $A_8+A_1$ is supported on two distinct families of Enriques surfaces:
once nodally primitive, the other time associated to the overlattice $E_8+A_1$ (Kond\=o's family of type I).
In accordance with Theorem \ref{thm} \textit{(ii)},
we claim that this situation is not far from optimal:

\begin{Proposition}
\label{prop:number}
There are exactly two maximal root types which are
supported on three distinct one-dimensional families of Enriques surfaces:
\[
A_7+2A_1,\;\;\; 3A_3.
\]
All other moduli spaces of Enriques surfaces have either one or two components.
\end{Proposition}

Note that Proposition \ref{prop:number} directly gives Theorem \ref{thm} \textit{(ii)},
thus completing the proof of Theorem \ref{thm} in view of \ref{ss:summary}.

\subsection{Outline of the proof of Proposition \ref{prop:number}}

To prove Proposition \ref{prop:number},
we shall study each root type separately,
say $R$ supported on an Enriques surface $Y$.
Here is an outline of the overall strategy, compare \ref{ss2:A_9}, \ref{ss:not}.
We start with an isotropic vector $E$ as in Table \ref{T1}
such that the perpendicular root lattice $R_0$ has rank $8$.
By Proposition \ref{prop:base}, $\mathop{\rm Jac}(Y)$ is an extremal rational elliptic  surface,
and $Y$ arises from $\mathop{\rm Jac}(Y)$ by the base change technique.
Using \cite{MP}, we can single out all extremal rational elliptic surfaces
admitting an embedding of $R_0$ into their singular fibers.
In fact, there may be quite a few fiber configurations,
but then the complimentary conditions kick in:
by Proposition \ref{prop:E} (and the way we set everything up),
the isotropic vector $E$ equips $Y$ with a smooth rational bisection $B$
which meets the nodally primitive closure $R_0'$ of $R_0$ in a prescribed way by Lemma \ref{lem:bi}.
By Corollary \ref{cor:fibers},
if $R_0$ is not nodally primitive, neither is $R$, so necessarily such a case is supported
on Enriques surfaces $Y$ of root type some overlattice of $R$,
determined as above.
The crucial problem then is whether $Y$ admits a smooth rational curve $C$,
 appearing as a multisection,
which supplements the root type $R_0$ supported on the fibers
to the root type $R$ which we are aiming for.
If the bisection $B$ does not suffice to this end,
we can in fact often rule out the existence of $C$ by checking
which intersection numbers with the given smooth rational curves on $Y$
are compatible with the fact that $\mathop{\rm Num}(Y)$ has rank 10, compare
\ref{sss-4}.

As an illustration, recall that for root type $A_9$, we have derived an irreducible moduli space in \ref{ss2:A_9},
while for $A_8+A_1$, there were two moduli components determined
in \ref{ss:not}.

In what follows, we shall not go into the details of the proof of Proposition \ref{prop:number}
and only sketch the arguments for the two root types with three moduli components.
The paper then concludes with an overview of the root types with exactly two moduli components
(such as $A_8+A_1$).

\subsection{Root type $A_7+2A_1$}
\label{ss:A7+2A1}

For the root type $R=A_7+2A_1$,
we have already seen two components of the moduli space:
on the one hand clearly the one displayed in Table \ref{T2} under $R$;
on the other hand related to the root type $D_6+3A_1$ in Figure \ref{fig:D6+3A1},
compare Remark \ref{rem:prim}.
Furthermore, it is supported on
Kond\=o's family of type I,
i.e.~the family of root type $E_8+A_1$
as one can easily locate the root type $A_7+2A_1$
in the graph of the 16 $(-2)$-curves on these Enriques surfaces from \cite[Fig.~1.4]{Kondo-Enriques}.
We claim that these three families are distinct,
and that they comprise all Enriques surfaces supporting the root type $R$.

To exhibit a complete argument,
it turns out to be more convenient to
return to the isotropic vector
\[
E = (0,0,a_1^\vee)+h
\]
from Table \ref{T1} (as opposed to the one used in \ref{sss:A_7+2A_1}).
Perpendicular, we find the root lattice $R_0=A_7+A_1\subset R$
with proper even overlattices
\[
E_7+A_1
E_8,
\]
always perpendicular to the bisection $a_1$ coming from the remaining orthogonal summand $A_1$.
We continue by working out the possible fiber configurations,
depending on which of the above overlattices is supported on $(-2)$-curves on $Y$.

\subsubsection{$\mathop{\rm Jac}(Y)=X_{8211}$}
\label{sss-1}

If $R$ is nodally primitive,
then $R_0$ embeds into singular fibers $I_8,I_2$
by Corollary \ref{cor:fibers},
compatibly with the bisection $a_1$ meeting both fibers in a single component by Lemma \ref{lem:bi}.
That is, $\mathop{\rm Jac}(Y)=X_{8211}$ with quadratic twist and section of height $2$
explicitly computable.
Recall from \ref{sss:A_7+2A_1} that this leads to two parametrisations
which we discuss in the sequel.

\subsubsection{Square parametrization}
\label{sss-1a}

Following \ref{sss:2t}, we first assume that the section $P'$ of height $2$ on the quadratic twist $X'$ has
$x$-coordinate $x(P')$ a square.
Solving for such a section directly leads to a one-dimensional family of K3 surfaces $\mathcal X_1'$ with
\[
x(P') = -16(t-\lambda)^2/(\lambda^2+4)^2,\;\;\;
q=4(\lambda^2+8)t^2-4\lambda(\lambda^2+12)t-(\lambda^2-4)^2
\]
On the family of K3 covers $\mathcal X_1$,
the induced section $P$ generates a lattice $L$ of rank $19$ together with the torsion sections and the
components of the reducible fibers (generically twice $I_8$, twice $I_2$).
We claim that generically
\[
\mathop{\rm NS}(\mathcal X_1)=L.
\]
Equivalently, $P$ and the torsion sections generate $\mathop{\rm MW}(\mathcal X_1)$.
To see this, we compute
the discriminant of $L$ as $64$ by \cite[\S 11 (22)]{SSh}.
Hence there could only be some hidden two-divisibility among the given sections.
This can be ruled out explicitly by checking $P$ itself and the sum of $P$ and either four-torsion section,
thus verifying the claim.

\subsubsection{Non-square parametrisation}
\label{sss-1b}

The second case from \ref{sss:2t} assumes that $x(P')$ is a non-square.
Thus it encodes precisely the quadratic twist.
Solving for such a section, we find a second family of Enriques surfaces
supporting the root type $R$.
It is encoded in the family of quadratic twists $\mathcal X_2'$ of $X_{8211}$ with infinite section $P'$ with
\[
x(P') = (t^2+2s+2)/(s^2-1).
\]
The crucial difference to the previous case is that $P'$ meets both $I_0^*$ fibers at the same component as the zero section $(0,0)$.
This subtlety facilitates that the sum $P'+(0,0)$ becomes two-divisible in $\mathop{\rm MW}(\mathcal X_2')$;
that is,
\[
P'+(0,0) = 2Q'
\]
for a section $Q'\in\mathop{\rm MW}(\mathcal X_2')$ of height $1/2$ meeting the $I_8$ fiber at $\Theta_2$
(components numbered cyclically starting from the identity component $\Theta_0$)
and both $I_0^*$ fibers non-trivially.
On the family of K3 covers $\mathcal X_2$,
we derive a section $Q$ of height $1$
which is easily checked not to allow for further divisibilities in $\mathop{\rm MW}(\mathcal X_2)$.
Hence $\mathop{\rm NS}(\mathcal X_2)$ has generically rank $19$ and discriminant $16$, again by  \cite[\S 11 (22)]{SSh}.
In particular, $\mathcal X_2$ is genuinely different from $\mathcal X_1$,
and so are their Enriques quotients.

In fact, we claim that the family of Enriques surfaces
arising from $\mathcal X_2$ through the base change type involution $\imath$
connected to $P$ as in Section \ref{s:tech}
supports the root type $E_7+2A_1$.
To see this, one checks using the height pairing from Mordell-Weil lattices \cite{ShMW}
that $Q$ is not only disjoint from the zero section $O$, but also from $P$.
Adding its image under $\pi$
to the diagram of $(-2)$-curves on the Enriques quotients
given by fiber components and bisection $a_1$ induced by $P$,
we depict the claimed root type $E_7+2A_1$ in Figure \ref{fig:A7+2A1}.

\begin{figure}[ht!]
\setlength{\unitlength}{1mm}
\begin{picture}(60,35)(0,-5)

\multiput(0,10)(10,0){2}{\circle*{1.5}}
\multiput(30,10)(10,0){4}{\circle*{1.5}}

\multiput(10,0)(10,0){3}{\circle*{1.5}}
\multiput(10,20)(10,0){3}{\circle*{1.5}}

\put(30,9.6){\line(1,0){30}}
\put(30,10.4){\line(1,0){30}}

\put(10,0){\line(1,0){20}}
\put(10,20){\line(1,0){20}}

\put(10,0){\line(0,1){20}}
\put(30,20){\line(0,-1){20}}

\qbezier(0,10)(10,-10)(20,0)
\qbezier(0,10)(10,30)(20,20)
\qbezier(0,10)(10,-25)(50,10)
\qbezier(0,10)(10,45)(50,10)

\put(-9,7){$\pi(Q)$}
\put(37,6){$a_1$}
\put(60,6){$A_1$}

\put(31,17){$e_1$}
\put(21,17){$e_4$}
\put(11,17){$e_3$}
\put(11,7){$e_2$}

\put(1.5,9){$e_5$}
\put(20,2){$e_6$}
\put(31,1){$e_7$}

\end{picture}
\caption{Root type $E_7+2A_1$ supported on $\mathcal X_2/\imath$}
\label{fig:A7+2A1}
\end{figure}

\subsubsection{Multiple $I_8$ fiber}
\label{sss-2}

Still assuming that $R_0$ is nodally primitive,
we can finally impose that the $I_8$ fiber has multiplicity two.
This implies that the bisection splits into zero section and two-torsion section on the K3 cover.
Thus we find another one-dimensional family of K3 surfaces
which in fact not only supports the root type $A_7+2A_1$, but also $E_8+A_1$.
That is, we end up with Kond\=o's type I family of Enriques surfaces with finite automorphism group.
Generically, it has discriminant $4$.

\subsubsection{Overlattices of $R_0$}
\label{sss-4}

If the nodal primitive closure of $R_0$ were to be $E_7+A_1$ or $E_8$,
then necessarily $R$ would have nodal primitive closure $E_7+2A_1$ or $E_8+A_1$,
so these cases are covered by \ref{sss-1b}, \ref{sss-2}.
With a view towards Proposition \ref{prop:number},
this concludes the analysis of the elliptic fibrations possibly induced by the isotropic vector $E$.

For demonstration reasons, however, we decided to include an argument
to rule out that $R_0'=E_8$.
This nodal primitive closure of $R_0$ would
correspond to a fiber of Kodaira type $II^*$,
thus $\mathop{\rm Jac}(Y)=X_{211}$ or $X_{22}$,
and $A_7+A_1$ would embed into $\tilde E_8$ as
\[
A_1=\mathbb{Z} e_2 \;\;\; \text{ and } A_7=\langle e_1, e_4, e_5, \hdots,e_9\rangle
\]
where $e_9$ denotes the simple fiber component of the $II^*$ fiber.
Thus the remaining orthogonal summand $A_1$ from $R$ would have to correspond to a smooth rational curve $C$
serving as a multisection $C$ meeting only the fiber component $e_3$.
Note that since $C$ splits into two disjoint multisections on the K3 cover,
the intersection multiplicity is even:
\[
C.e_2=2n, \;\;\; n\in\N.
\]
Setting $C.a_1=o\in\N_0$ for the bisection $a_1$, we can solve for the resulting $(11\times 11)$-Gram matrix to have rank ten;
equivalently
\[
12n^2-4no-1=0.
\]
Reducing modulo $2$, we deduce that there are no integral solutions.
Hence the above configuration of $(-2)$-curves cannot occur on any Enriques surface.

\begin{Remark}
We point out a crucial subtlety:
the root type $R=A_7+2A_1$ is supported on the Enriques surfaces
for root type $E_8+A_1$ by \ref{sss-2},
but not in such a way that the sublattice $A_7+A_1$ is supported on a divisor of Kodaira type $II^*$.
This conforms with the finite graph of smooth rational curves on such Enriques surfaces
from \cite[Fig.~1.4]{Kondo-Enriques}.
\end{Remark}

\subsubsection{Conclusion}

Arguing with elliptic fibrations induced by the isotropic vector
$
E = (0,0,a_1^\vee)+h
$,
we have found exactly three families of Enriques surfaces
supporting the root type $R=A_7+2A_1$.
Since the discriminants of the covering K3 surfaces are all distinct by \ref{sss-1a} - \ref{sss-2},
these families correspond to three distinct moduli spaces.
Hence Proposition \ref{prop:number} for $R=A_7+2A_1$ follows.
\hfill \ensuremath{\Box}

\subsection{Root type $3A_3$}
\label{ss:3A3}

The root type $R=3A_3$ behaves quite similar to the previous one.
Especially, this appeals to the fact
that the isotropic vector $E=(a_2^\vee,0,0)+h/2$
yields the same perpendicular root lattice $R_0=2A_3+2A_1$
as in \ref{sss:A_7+2A_1} (although we opted to argue with a different isotropic vector in \ref{ss:A7+2A1}).
We follow the same steps,
based on the elliptic fibration induced by $E$ with smooth rational bisection $a_2$ from the third $A_3$ summand.

\subsubsection{$\mathop{\rm Jac}(Y)=X_{4422}$}
\label{sss-11}

If $R_0$ is nodally primitive, then $R_0$ embeds into singular fibers $I_4, I_4, I_2, I_2$ by Lemma \ref{prop:fibers}.
Thus~$\mathop{\rm Jac}(Y)=X_{4422}$ by \cite{MP},
and one computes the quadratic twist and section displayed in Table \ref{T2}.
Generically, $a_2$ splits into zero section $O$ and section $P$ of height $2$ on the K3 cover $X$.
One can check that $P$ together with the torsion sections generically generates $\mathop{\rm MW}(X)$;
hence $\mathop{\rm NS}(X)$ generically has discriminant $128$ by \cite[\S 11 (22)]{SSh}.

\subsubsection{Multiple $I_4$ fiber}
\label{sss-22}

What's special in the above set-up is that imposing one of the $I_4$ fibers to ramify
causes the section $P$ to become $2$-torsion.
That is, this gives a second family which in fact supports the root type $D_6+A_3$
as listed in Table \ref{T2}.
Generically, the K3 cover $X$ has discriminant $32$ by \cite[\S 11 (22)]{SSh},
since $\mathop{\rm MW}(X) = \mathop{\rm MW}(S)\cong \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.

\subsubsection{Overlattices of $R_0$}

It remains to study the overlattices of $R_0$
which could be supported on $(-2)$-curves on the Enriques surface $Y$.
There are five candidates:
\[
D_5+A_3, D_6+2A_1, E_7+A_1, D_8, E_8.
\]
By Lemma \ref{prop:fibers},
the singular fibers of the elliptic fibration arising from $E$
are then exactly given by the extended Dynkin diagrams of the orthogonal summands in $R_0'$.
First  we have to decide which fiber configurations  support the original root type $R_0$ on fiber components.
This rules out $R_0'=D_8$ and $ E_8$ right away.
Similarly, if $R_0'$ were to equal $E_7+A_1$,
then the glue vector for $2A_3+A_1\hookrightarrow E_7$ involving $A_1^\vee$ would not meet $a_2$ integrally, contradiction.
A little more involved, if $R_0'=D_6+2A_1$,
then supporting $R_0$ would require a smooth rational multisection $C$
meeting the $I_2^*$ fiber at exactly the middle component $d_3$ (with even multiplicity as in \ref{sss-4})
while each $I_2$ fiber is met at one component with multiplicity one.
Solving for the resulting Gram matrix to have rank ten,
one easily finds that there cannot be such a multisection.

\subsubsection{$\mathop{\rm Jac}(Y)=X_{141}$}
\label{sss-33}

Finally we analyse the case $R_0'=D_5+A_3$.
Here $\mathop{\rm Jac}(Y)=X_{141}$,
and we embed  one $A_3+2A_1$ into a single fiber of type $I_1^*$,
and the perpendicular $A_3$ into a fiber of type $I_4$.
Enriching $R$ by the additional $(-2)$-curve from $R_0'$,
one calculates
that the nodally primitive closure of $R$ contains $D_6+A_3$.
Here the original glue vector $(0,0,a_2^\vee)+h/2$
corresponds to $E=(d_1^\vee,0)+h/2$,
and $D_5$ embeds as $\langle d_2,\hdots, d_6\rangle$ into $E^\perp\cap R$.
By Lemma \ref{lem:bi},
the smooth rational curve $d_1$ thus gives a bisection
meeting the $I_1^*$ fiber in the close fiber components $d_2$ and $d_0$, say.
If the $I_4$ fiber is not ramified,
then we obtain exactly the family for root type $D_6+A_3$ from Table \ref{T2}.
It indeed supports the root type $3A_3$ as indicated in the following figure.

\begin{figure}[ht!]
\setlength{\unitlength}{.6mm}
\begin{picture}(120,35)(0,0)

\multiput(0,0)(50,0){2}{\circle*{1.5}}

\multiput(0,30)(50,0){2}{\circle*{1.5}}

\multiput(15,15)(20,0){2}{\circle*{1.5}}

\multiput(65,15)(20,0){2}{\circle*{1.5}}

\multiput(100,0)(0,30){2}{\circle*{1.5}}

\put(115,15){\circle*{1.5}}

\multiput(0,0)(50,0){3}{\line(1,1){15}}

\multiput(0,30)(50,0){3}{\line(1,-1){15}}

\multiput(50,0)(50,0){2}{\line(-1,1){15}}

\multiput(50,30)(50,0){2}{\line(-1,-1){15}}

\put(15,15){\line(1,0){20}}

\put(65,14.5){\line(1,0){20}}

\put(65,15.5){\line(1,0){20}}

\put(2,14){$A_3$}
\put(102,14){$A_3$}

\put(52,-5){$d_0$}

\put(52,32){$d_2$}

\put(64,8){$d_1$}

\end{picture}
\end{figure}

We emphasise that the given family is equivalent to the one from  \ref{sss-22}.
Indeed, the linear system of twice the central  divisor $d_0+d_1+d_2+d_3$ of Kodaira type $I_4$ in the previous graph of $(-2)$-curves on $Y$
exactly leads  to the fibration from  \ref{sss-22}.

\subsubsection{Multiple $I_4$ fiber (again)}

Once more we may assume that the $I_4$
fiber of $Y$ has multiplicity two.
Then the section which $a_2$ splits off on the K3-cover $X$ becomes two-torsion,
and generically $X$ has discriminant $8$ by  \cite[\S 11 (22)]{SSh}.
Indeed, $Y$ supports the root type $D_9$
(albeit a different model compared to Table \ref{T2}),
and the family coincides with Kond\=o's type II from \cite{Kondo-Enriques}.

\subsubsection{Dinstinct moduli components}

We have already checked that the three one-dimensional families of Enriques surfaces
have K3 covers of Picard number $19$ and generically different discriminant,
hence they are genuinely distinct.
This perfectly agrees with the fact
that the families support an increasing chain of root lattices
\[
3A_3\subset D_6+A_3\subset D_9.
\]
This completes the proof of Proposition \ref{prop:number} for root type $3A_3$.
\hfill \ensuremath{\Box}

\begin{Remark}
We also deduce right away that the root type $D_6+A_3$
is supported on at least two moduli components.
Arguing with the isotropic vector from Table \ref{T1} and $R_0=D_6+2A_1$,
one  finds along the same lines as above that there are exactly two components.
\end{Remark}

\subsection{Root types with two moduli components}

We conclude this paper by listing the root types from Theorem \ref{thm:31}
which are supported on exactly two distinct families of Enriques surfaces.
In detail, we include the original root type $R$,
the perpendicular complement $R_0$ of the isotropic vector (from Table \ref{T1} resp.~\ref{ss:not})
and for either moduli component the singular fibers (with multiplicity) into which $R_0$ embeds.
We also indicate the overlattice $R'$ supported on the second component.

\begin{table}[ht!]
$$
\begin{array}{|ccc|cc|}
\hline
R & R_0& \text{fibers} & R' & \text{fibers}\\
\hline
A_8+A_1 & A_8 & I_9 & E_8+A_1 & II^*\\
A_7 + A_2 & A_5+A_2+A_1 & I_6,I_3,I_2 & E_7 + A_2 & 2I_6, I_3, I_2\\
A_5+A_3+A_1 & 2A_3+2A_1 & I_4,I_4,I_2,I_2 & E_6+A_3 & I_1^*, I_4\\
A_5+2A_2 & A_5+A_2+A_1 & I_6,I_3,I_2 & E_7+A_2 & 2I_6,I_3,I_2\\
D_8+A_1 & A_7+A_1 & I_8, I_2 & E_8+A_1 & 2I_8, I_2
\\
D_6+A_3 & D_6+2A_1& I_2^*, I_2,I_2 & D_9 & I_4^*
\\
D_4+A_3+2A_1 & 2A_3+2A_1& I_4,I_4,I_2,I_2
& D_7+2A_1 & 2I_4,I_4,I_2,I_2
\\
\hline
\end{array}
$$
\end{table}

The calculations follow precisely the lines of \ref{ss:A7+2A1}, \ref{ss:3A3}
and are therefore omitted for brevity.

\subsection*{Acknowledgements}

Thanks to S\l awomir Rams and Chris Peters for  discussions on the subject
and to JongHae Keum for bringing this problem to my attention and helpful comments.