Enumerating Cuspidal Curves on Toric Surfaces

TL;DR

This paper counts rational cuspidal curves with nodes on toric surfaces using tropical geometry, providing a combinatorial approach to a classical enumerative problem in algebraic geometry.

Contribution

It introduces a method to count such curves via tropicalization and patchworking, linking algebraic and tropical enumerative geometry.

Findings

Number of cuspidal curves equals tropical counts with multiplicities

Classification of tropical limits of algebraic curves

Application of patchworking to reconstruct algebraic families

Abstract

Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by fixing some algebraic or geometric invariants (such as degree, genus and types of singularities), the problem usually takes the form of "how many curves of that class pass through a configuration of n points in general position?" Tropical Geometry deals with certain piecewise-linear complexes, which arise as degeneration of families of complex algebraic varieties, and can also be described algebraically using "max-plus" algebra, (The tropical semi-field). The problem we solve is that of counting rational curves with one cusp and certain number of nodes on toric surfaces, passing through a configuration of sufficient points in general position. We show that that this number equals the number of certain tropical curves counted with multiplicities and we describe these curves and their multiplicities. The main tools are tropicalization and patchworking. In tropicalization we pass from an equisingular family of curves to a special limit fiber which can be described in terms of tropical data and analytic data. We then classify these possible limits, and use the patchworking theorem to reconstruct the families that correspond to them.