A Fock Space approach to Severi Degrees of Hirzebruch Surfaces

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.