A Fock space approach to Severi degrees

TL;DR

This paper introduces a Fock space framework to compute Severi degrees for certain surfaces, linking algebraic geometry with operator formalism and providing explicit formulas for various cases.

Contribution

It develops a novel Fock space approach to express Severi degrees as matrix elements of an exponential operator, unifying different enumerative problems.

Findings

Explicit matrix element formulas for Severi degrees of CP1 x CP1.

Exact eigenvalue formulas for genus 1 invariants of E x CP1.

Determination of Severi degrees of CP2 using disk multiple cover formulas.

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.