Modular subvarieties and birational geometry of $SU_C(r)$

TL;DR

This paper investigates the birational geometry of moduli spaces of semi-stable vector bundles on algebraic curves, revealing their structure as fibrations over symmetric products with fibers as GIT quotients, and connects to classical modular varieties.

Contribution

It establishes the birational equivalence of these moduli spaces to fibrations over symmetric products with GIT quotient fibers, linking to classical modular varieties in low-rank cases.

Findings

Moduli space $U_C(r,0)$ is birational to a fibration over $C^{(rg)}$ with GIT quotient fibers.

The subspace $SU_C(r)$ is birational to a fibration over $P^{(r-1)g}$ with similar fibers.

In low rank and genus, the construction yields classical modular varieties within Coble hypersurfaces.

Abstract

Let $C$ be an algebraic smooth complex genus $g>1$ curve. The object of this paper is the study of the birational structure of the coarse moduli space $U_C(r,0)$ of semi-stable rank r vector bundles on $C$ with degree 0 determinant and of its moduli subspace $SU_C(r)$ given by the vector bundles with trivial determinant. Notably we prove that $U_C(r,0)$ (resp. $SU_C(r)$) is birational to a fibration over the symmetric product $C^(rg)$ (resp. over $P^{(r-1)g}$) whose fibres are GIT quotients $(P^{r-1})^{rg}//PGL(r)$. In the cases of low rank and genus our construction produces families of classical modular varieties contained in the Coble hypersurfaces.