The $C_2$-equivariant ordinary cohomology of complex quadrics III: Exceptional cases

TL;DR

This paper completes the calculation of $C_2$-equivariant cohomology for complex quadrics with multiple fixed components, and applies these results to refine classical enumerative geometry facts such as lines on a cubic surface.

Contribution

It provides the final computations for equivariant cohomology of symmetric quadrics with complex fixed sets and applies these to refine classical geometric results.

Findings

Completed $C_2$-equivariant cohomology calculations for complex quadrics with multiple fixed components.

Derived an equivariant refinement of the 27 lines on a cubic surface.

Extended the understanding of equivariant topology in algebraic geometry.

Abstract

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components. These calculations imply one for a $C_2$-equivariant Grassmannian, which we use to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.