A Fock Space approach to Severi Degrees of Hirzebruch Surfaces

Abstract

The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through some general points in a surface. In this paper we study Severi degrees as well as several types of Gromov-Witten invariants of the Hirzebruch surfaces $F_k$, and the relationship between these numbers. To each Hirzebruch surface $F_k$ we associate an operator $\mathsf{M}_{F_k} \in \mathcal{H}[\mathbb{P}^1]$ acting on the Fock space $\mathcal{F}[\mathbb{P}^1]$. Generating functions for each of the curve-counting theories we study here on $F_k$ can be expressed in terms of the exponential of the single operator $\mathsf{M}_{F_k}$, and counts on $\mathbb{P}^2$ can be expressed in terms of the exponential of $\mathsf{M}_{F_1}$. Several previous results can be recovered in this framework, including the recursion of Caporaso and Harris for enumerative curve counting on $\mathbb{P}^2$, the generalization by Vakil to $F_k$, and the relationship of Abramovich-Bertram between the enumerative curve counts on $F_0$ and $F_2$. We prove an analog of Abramovich-Bertram for $F_1$ and $F_3$. We also obtain two differential equations satisfied by generating functions of relative Gromov-Witten invariants on $F_k$. One of these recovers the differential equation of Getzler and Vakil.