A Fock space approach to Severi degrees

Abstract

The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface. We express the Severi degrees of CP1 x CP1 as matrix elements of the exponential of a single operator M on Fock space. The formalism puts Severi degrees on a similar footing as the more developed study of Hurwitz numbers of coverings of curves. The pure genus 1 invariants of the product E x CP1 (with E an elliptic curve) are solved via an exact formula for the eigenvalues of M to initial order. The Severi degrees of CP2 are also determined by M via the (-1)^(d-1)/d^2 disk multiple cover formula for Calabi-Yau 3-fold geometries.

20 pages, 6 figures. Revised in response to referee comments. To appear in Proceedings of the London Mathematical Society

References in corpus (2)

• Floor decompositions of tropical curves : the planar case
• Topological recursion relations and Gromov-Witten invariants in higher genus

Cited by in corpus (5)

• Refined floor diagrams from higher genera and lambda classes
• Enumerative geometry of elliptic curves on toric surfaces
• The Hurwitz space of covers of an elliptic curve $E$ and the Severi variety of curves in $E\times {\mathbb{P}}^1$
• A Fock Space approach to Severi Degrees of Hirzebruch Surfaces
• Counting singular curves with tangencies