Compactified Jacobians of curves with spine decompositions

TL;DR

This paper compares two types of compactified Jacobians for algebraic curves, establishing conditions under which they coincide, and proves the projectivity of the fine moduli spaces of P-quasistable sheaves.

Contribution

It provides a sufficient condition for the equality of two compactification schemes of Jacobians and proves the projectivity of the fine moduli spaces.

Findings

The fine moduli spaces are shown to be projective.

A sufficient condition for the two types of moduli spaces to be equal.

Comparison between P-quasistable and S-equivalence class moduli schemes.

Abstract

A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri's moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Here we give a sufficient condition for when these two types of moduli spaces are equal.