Title: Curve counts and quadratic forms

Abstract:

An old problem of enumerative geometry concerns how many rational curves of degree d pass through 3d-1 points in the plane. Over the complex numbers, the answer does not depend on the choice of points in the plane so long as they are generic. However, over non-algebraically closed fields, this is no longer the case. I will discuss a framework (joint work with Kass, Levine and Wickelgren) within which one can define invariant counts of rational curves passing through points in the plane (or a del Pezzo surface) over a perfect field of characteristic not 2 or 3. The count is no longer a number, but a quadratic form over the given field. Over the complex numbers, the rank of the quadratic form recovers the usual count. Over the real numbers, the signature of the quadratic form recovers Welschinger's signed count. Over other fields, each invariant of quadratic forms (discriminant, Hasse-Witt,...) gives information about rational curves over that field.

In another direction, mirror symmetry relates counts of rational curves on a toric Fano variety over the complex numbers to the Jacobian ring of a Laurent polynomial. A categorification of the Jacobian ring is given by the category of matrix factorizations. I will discuss a framework (joint work with Sela) to extract counts of rational curves over the real numbers from matrix factorizations equipped with non-Archimedean norms. These matrix factorizations are constructed from Clifford algebras. The Calabi-Yau structure on the category of matrix factorizations plays a crucial role.

References:
A quadratically enriched count of rational curves (with Kass, Levine and Wickelgren)

https://arxiv.org/abs/2307.01936

A relative orientation for the moduli space of stable maps to a del Pezzo surface (with Kass, Levine and Wickelgren)

https://arxiv.org/abs/2307.01941

Numerical invariants of normed matrix factorizations (with Sela)

https://arxiv.org/abs/2412.04437