On GIT quotients of Hilbert and Chow schemes of curves

TL;DR

This paper discusses new results on the geometric invariant theory (GIT) quotients of Hilbert and Chow schemes of algebraic curves, extending previous work and providing a new compactification of the universal Jacobian.

Contribution

It extends previous GIT results for curves in projective space and offers a complete description of GIT quotients in a specific degree range, introducing a new compactification of the universal Jacobian.

Findings

Extended GIT results up to degree d > 4(2g-2).

Complete description of GIT quotient for 2(2g-2)<d<7/2(2g-2).

Introduced a new compactification of the universal Jacobian.

Abstract

The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.