Abstract

Associated to any smooth projective curve $C$ is its degree $d$ Jacobian variety, parametrising isomorphism classes of degree $d$ line bundles on $C$. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves $\overline{\mathcal{M}}_{g,n}$, depending on the choice of a stability condition. In this talk I will begin by introducing these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). I will then introduce new analogues of these moduli stacks, where the sheaves being parametrised are of a fixed unstable Harder—Narasimhan type, and explain how non-reductive GIT allows one to prove the existence of quasi-projective moduli spaces for these stacks, which in many cases are naturally projective.   This talk is based on arXiv:2210.11457 and upcoming work.