Fourier transformation and the Abel-Jacobi section

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). Using the Fourier-Mukai transform, this locus can be expressed as a power of twisted theta divisor. 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 its class can be computed using Pixton’s formula on the universal double ramification cycle formula.

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 an explicit computation of Fourier transform on the compactified Jacobian 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.