“The double-double ramification cycle”
Wednesday, April 28, at 14:00 - Zoom - David Homes (University of Leiden)
A basic question in the geometry of Riemann surfaces is to decide when a given divisor of degree 0 is the divisor of a rational function (is principal). In the 19th century Abel and Jacobi gave a beautiful solution: one writes the divisor as the boundary of a 1-cycle, and the divisor is principal if and only if every holomorphic differential integrates to zero against this cycle.
From a modern perspective it is natural to allow the curve and divisor to vary in a family, perhaps allowing the curve to degenerate to a singular (stable) curve so that the corresponding moduli space is compact. The double ramification cycle can then be seen as a virtual fundamental class of the locus in the moduli space of curves over which our divisor becomes principal.
We will focus on two basic questions: where does the double ramification cycle naturally live, and what happens when we intersect two double ramification cycles? We will see why (logarithmically) blowing up the moduli space can make life easier. This is joint work with Rosa Schwarz, building on easlier work with Aaron Pixton and Johannes Schmitt.