Birational maps of moduli of Brill-Noether pairs

TL;DR

The paper constructs explicit birational maps between moduli spaces of stable pairs on a general curve, revealing new relationships and conjectures about their birational geometry for large stability parameters.

Contribution

It provides an explicit birational map between specific moduli spaces of stable pairs, advancing understanding of their birational relationships and proposing a broader conjecture.

Findings

Explicit birational map between $G_{\alpha}(n,d,n+1)$ and $G_{\alpha}(1,d,n+1)$

Evidence supporting a general birational correspondence between $G_{\alpha}(a,d,a+z)$ and $G_{\alpha}(z,d,a+z)$

Conjecture on birational equivalence for general curves with high genus

Abstract

Let $C$ be a smooth projective irreducible curve of genus $g$. And let $G_{\alpha}(n,d,l)$ be the moduli space of $\alpha$ stable pairs of a vector bundle of $\rank n, \deg d$ and a subspace of $H^0(C,E)$ of $\dim = l $. We find an explicit birational map from $G_{\alpha} (n, d, n+1)$ to $G_{\alpha} (1, d, n+1)$ for $C$ general, $1/\alpha \gg 0$ and $g \ge n^2-1$. Because of this and other examples, we conjecture $G_{\alpha} (a, d, a+z)$ maps birationally to $G_{\alpha} (z, d, a+z)$ for $1/ \alpha \gg 0$ and $C$ general with $g>2$.