On the cup product for Hilbert schemes of points in the plane

TL;DR

This paper analyzes the cohomology ring structure of Hilbert schemes of points in the plane, using combinatorial and geometric techniques to establish conditions for non-vanishing products of cohomology classes.

Contribution

It provides new combinatorial criteria for the non-vanishing of cup products in the cohomology of Hilbert schemes, connecting cellular decompositions with Białynicki-Birula theory.

Findings

Derived necessary conditions for non-vanishing cohomology class products.

Connected cellular decomposition with Białynicki-Birula stratification techniques.

Established combinatorial criteria for intersection properties of cell closures.

Abstract

We revisit Ellingsrud and Str{\o} mme's cellular decomposition of the Hilbert scheme of points in the projective plane. We study the product of cohomology classes defined by the closures of cells, deriving necessary conditions for the non-vanishing of cohomology classes. Though our conditions are formulated in purely combinatorial terms, the machinery for deriving them includes techniques from Bia{\l} ynicki-Birula theory: We study closures of Bia{\l} ynicki-Birula cells in complete complex varieties equipped with ample line bundles. We prove a necessary condition for two such closures to meet, and apply this criterion in our setting.