Coherent systems and modular subvarieties of SU_C(r)

TL;DR

This paper investigates the birational geometry of moduli spaces of vector bundles and coherent systems on algebraic curves, revealing their structure as fibrations over projective varieties and comparing various stability notions.

Contribution

It demonstrates that certain moduli spaces are birationally equivalent to fibrations with GIT quotient fibers and relates different stability concepts in this context.

Findings

Moduli spaces are birationally equivalent to fibrations over projective varieties.

Identifies relations between slope stability, α-stability, and GIT stability.

Constructs families of classical modular varieties within Coble hypersurfaces.

Abstract

Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients $(\PP^{r-1})^{rg}//PGL(r)$, where $r$ is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, $\alpha$-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when $C$ has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.