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

Steven R. Costenoble, Thomas Hudson

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 $\mathbb{P}^3$.

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

Abstract

In this, the last of three papers about -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 -equivariant Grassmannian, which we use to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in .

Paper Structure

This paper contains 9 sections, 20 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Quadrics of type (B,D) and (D,B)
  4. Quadrics of type (D,D)
  5. A Grassmannian
  6. 27 lines
  7. The even case
  8. The odd case
  9. The product of two projective lines

Key Result

Proposition 3.4

We have cofibration sequences

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 32 more