Title: Fourier transformation and the Abel-Jacobi section

Abstract: Let v1, ... , vn be a vector of integers that sum to zero. On the relative Jacobian over the moduli space of smooth genus g curves with n sections, the Abel-Jacobi section maps a marked curve (C, x1, ..., xn) to a line bundle O(v1.x1+ ... + vn.xn). By the work of Hain, this locus can be expressed as a power of twisted theta divisor. A similar question arises when the curve acquires nodal singularities. The relative Jacobian can be compactified via stable rank 1 torsion-free sheaves. After blowing up the base, the Abel-Jacobi section extends and the class of resolved Abel-Jacobi sections can be computed using Pixton's formula.

We connect the class of resolved Abel-Jacobi sections and the intersection theory on compactified Jacobians. In this talk, we propose a conjectural closed formula for the pushforward of monomials of divisor classes on compactified Jacobians. This conjecture is motivated by Arinkin's Fourier transform on the compactified Jacobian, logarithmic geometry of Abel-Jacobi section, and combinatorial properties of the Pixton's formula. We verify the conjecture over various open loci of the base. This is joint work in progress with Samouil Molcho and Aaron Pixton.