Infinitesimal structure of the pluricanonical double ramification locus

We prove that a formula for the ‘pluricanonical’ double ramification cycle proposed by Janda, Pandharipande, Pixton, Zvonkine, and the second-named author is in fact the class of a cycle constructed geometrically by the first-named author. Our proof proceeds by a detailed explicit analysis of the deformation theory of the double ramification cycle, both to first and to higher order.

EXTENDING THE DOUBLE RAMIFICATION CYCLE BY RESOLVING THE ABEL-JACOBI MAP

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

We describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).