Classifying fine compactified universal Jacobians

Abstract: We introduce the notion of a fine compactified Jacobian of a nodal curve, as an arbitrary compact open subspace of the moduli space of rank-1 torsion-free simple sheaves. We show that fine compactified Jacobians correspond to a certain combinatorial datum, which is obtained by only keeping track, for all sheaves, of (1) the locus where it fails to be locally free, and (2) its multidegree. This notion generalizes to flat families of curves, and so does its combinatorial counterpart. When the family is the universal family over the moduli space of curves, we have the following results: (a) in the absence of marked points, we can fully classify these combinatorial data and deduce that the only fine compactified universal Jacobians are the classical ones (which were constructed by Pandharipande and Simpson in the nineties) and (b) in the presence of marked points there are exotic (and new) examples that cannot be obtained as compactified universal Jacobians associated to a polarization. This is a joint work in progress with Jesse Kass.

algebraic geometry

Audience: researchers in the topic

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