A Fock space approach to Severi degrees

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 P 1 × P 1 as matrix elements of the exponential of a single operator M S 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 × P 1 (with E an elliptic curve) are solved via an exact formula for the eigenvalues of M S to initial order. The Severi degrees of P 2 are also determined by M S via the ( − 1 ) d − 1 / d 2 disk multiple cover formula for Calabi–Yau threefold geometries.