A Fock space approach to Severi degrees of Hirzebruch surfaces
Year: 2025
Type: article
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 F_k , and the relationship between these numbers. To each Hirzebruch surface F k F_k we associate an operator M F k ∈ H [ P 1 ] \mathsf {M}_{F_k} \in \mathcal {H}[\mathbb {P}^1] acting on the Fock space F [ P 1 ] \mathcal {F}[\mathbb {P}^1] . Generating functions for each of the curve-counting theories we study here on F k F_k can be expressed in terms of the exponential of the single operator M F k \mathsf {M}_{F_k} , and counts on P 2 \mathbb {P}^2 can be expressed in terms of the exponential of M F 1 \mathsf {M}_{F_1} . Several previous results can be recovered in this framework, including the recursion of Caporaso and Harris [Invent. Math. 131 (1998), pp. 345–392] for enumerative curve counting on P 2 \mathbb {P}^2 , the generalization by Caporaso and Harris and later Vakil [Manuscripta Math. 102 (2000), pp. 53–84] to F k
DOI: 10.1090/btran/46
Author: Yaim Cooper
Institution: Institute for Advanced Study
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,750
OpenAlex ID: W4408778648