Pluricanonical cycles and tropical covers

TL;DR

This paper introduces new numerical invariants derived from logarithmic intersection theory on pluricanonical double ramification cycles, revealing properties similar to double Hurwitz numbers and enabling efficient tropical curve counts.

Contribution

It develops a tropical geometric framework for pluricanonical cycles, generalizing double Hurwitz numbers with novel invariants and computational methods.

Findings

Invariants are piecewise polynomial in ramification data.

They can be computed via counts of tropical curves with modified balancing.

The invariants are matrix elements of operators on the Fock space.

Abstract

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.