Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On linear series with negative Brill-Noether number   Brill-Noether theory studies the existence and deformations of curves in
projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the
moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle
having at least $(r+1)$ independent global sections. The Brill-Noether theorem
asserts that the map $\mathcal{W}^r_{d,g} \rightarrow \mathcal{M}_g$ is
surjective with general fiber dimension given by the number $\rho = g -
(r+1)(g-d+r)$, under the hypothesis that $0 \leq \rho \leq g$. One may
naturally conjecture that for $\rho < 0$, this map is generically finite onto a
subvariety of codimension $-\rho$ in $\mathcal{M}_g$. This conjecture fails in
general, but seemingly only when $-\rho$ is large compared to $g$. This paper
proves that this conjecture does hold for at least one irreducible component of
$\mathcal{W}^r_{d,g}$, under the hypothesis that $0 < -\rho \leq \frac{r}{r+2}
g - 3r+3$. We conjecture that this result should hold for all $0 < -\rho \leq g
+ C$ for some constant $C$, and we give a purely combinatorial conjecture that
would imply this stronger result.

19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin