byu algebraic geometry seminar

I'll give a very broad overview of Beauville's proof of the Yau—Zaslow formula for counting rational curves of genus g on a K3 surface. The goal is to get you excited about K3 surfaces and some of the topics we might cover along the way.

Section 1.1 of these notes on projective varieties.

Section 1.1 of these notes on projective varieties.

Section 5 of these notes on sheaves.

Sections 3 and 13 of these notes on algebraic geometry.

Section 15 of these notes on algebraic geometry.

Section 16 of these notes on algebraic geometry.

This talk will be a friendly introduction to using topological invariants in enumerative geometry and how one might use equivariant homotopy theory to answer enumerative questions under the presence of a finite group action. Recent work with Kirsten Wickelgren (Duke) defines a global and local degree in stable equivariant homotopy theory that can be used to compute the equivariant Euler characteristic and Euler number. I will discuss an application to counting orbits of rational plane cubics through an invariant set of 8 points in general position under a finite group action on $\mathbb{C}\mathbb{P}^2$, valued in the representation ring and Burnside ring. This recovers a signed count of real rational cubics when $\mathbb{Z}/2$ acts on $\mathbb{C}\mathbb{P}^2$ by complex conjugation.