On rational maps between moduli spaces of curves and of vector bundles

TL;DR

This paper explores the structure of the moduli space of rank 2 semistable vector bundles over algebraic curves, revealing a connection to moduli spaces of pointed genus zero curves through rational normal curves.

Contribution

It constructs large families of pointed rational normal curves within moduli spaces and interprets these in terms of moduli of genus zero curves, establishing new geometric links.

Findings

Existence of natural linear maps from SU_C(2) to projective space with modular meaning

Fibers of these maps are birational to M_{0,2g}

For g<4, fibers are isomorphic to GIT compactification M^{GIT}_{0,2g}

Abstract

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.