Birational maps of moduli of Brill-Noether pairs

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 $g>2$.