Weekly Bulletin

It is well-understood that Gromov-Witten (GW) invariants often fail to be enumerative. For example, when r is at least 3, the higher-genus GW invariants of P^r fail to count smooth curves in projective space in any transparent sense. The situation seems to be better when one fixes the complex structure of the domain curve. It was originally speculated that if X is a Fano variety, then the "fixed-domain" GW count of curves of sufficiently large degree passing through the maximal number of general points is enumerative. I will discuss some positive and negative results in this direction, focusing on the case of hypersurfaces. The most recent results are joint with Roya Beheshti, Brian Lehmann, Eric Riedl, Jason Starr, and Sho Tanimoto, and build on earlier work with Rahul Pandharipande and Alessio Cela.

After the work of Beauville and Deninger-Murre, Fourier transformation plays an important role to study the Chow group of abelian schemes. Arinkin extended the Fourier transformation to the relative compactified Jacobian of a family of integral local planar curves. In this talk, I will explain how Arinin’s Fourier transform can be used to compute the class of Abel-Jacobi sections on the relative Jacobian over the moduli space of integral nodal curves. This result partially recovers Pixton’s formula of Abel-Jacobi section proven by Bae-Holmes-Pandharipande-Schmitt-Schwarz. This is a joint work in progress with S. Molcho.