Plane Curves, Node Polynomials, and Floor Diagrams.

Abstract

This thesis is about utilizing and extending a combinatorial approach - based on tropical geometry - to the Gromov-Witten theory of the complex projective plane. It is comprised of three related projects.

The first project is concerned with the degrees of Severi varieties, which are given (beyond some threshold value) by "node polynomials." We compute several new cases of these polynomials as well as new leading coefficients for the general case. We also obtain an improved polynomiality threshold.

In the second project, we prove that the degrees of generalized Severi varieties are, above an explicit threshold, given by relative node polynomials. As in the first project, we compute the first several polynomials and determine the first few leading terms.

In the third project, joint with A. Gathmann and H. Markwig, we study descendant Gromov-Witten invariants and their relative analogues. We show that combinatorial gadgets called Psi-floor diagrams enumerate these invariants, thus establishing a new tropical correspondence theorem.