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A Quadratically Enriched Count Of Lines In Smooth Del Pezzo Surfaces Of Degree 2 And 4

Victor Chachay

Abstract

We give a computation of some Euler classes in Chow-Witt groups associated to the count of lines of smooth del Pezzo surfaces of degree 2 and 4. The description of Chow-Witt groups of projective bundles over Grassmannians for vector bundles that are not relatively orientable is the main part of the article. We show that the quadratic count is not enriched as the Chow-Witt group is isomorphic to the Chow group. In this setting, we give an expression of the classes of even rank in the Chow-Witt group as multiples of the hyperbolic element h. A direct application of this construction is the count for the del Pezzo surfaces.

A Quadratically Enriched Count Of Lines In Smooth Del Pezzo Surfaces Of Degree 2 And 4

Abstract

We give a computation of some Euler classes in Chow-Witt groups associated to the count of lines of smooth del Pezzo surfaces of degree 2 and 4. The description of Chow-Witt groups of projective bundles over Grassmannians for vector bundles that are not relatively orientable is the main part of the article. We show that the quadratic count is not enriched as the Chow-Witt group is isomorphic to the Chow group. In this setting, we give an expression of the classes of even rank in the Chow-Witt group as multiples of the hyperbolic element h. A direct application of this construction is the count for the del Pezzo surfaces.
Paper Structure (18 sections, 43 theorems, 58 equations)

This paper contains 18 sections, 43 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Motivic setup
  3. Stable homotopy and six functors
  4. Universal Euler class
  5. Homologies, cohomologies
  6. Orientations
  7. Euler class computation
  8. Definitions of the $K-$theory rings
  9. Relations between $K-$theory rings
  10. Cohomology and Rost-Schmid complexes
  11. Chow-Witt Euler classes computation
  12. Lines in a degree 2 del Pezzo
  13. Del Pezzo surfaces of degree 2
  14. Lines in a projective bundle
  15. Euler class of an odd rank sheaf
  16. ...and 3 more sections

Key Result

Theorem 1.3

A quadratic enrichment of the 56 lines in a del Pezzo surface of degree 2 is:

Theorems & Definitions (117)

  • Definition 1.1: Relative orientability
  • Theorem 1.3: Main Theorem \ref{['THO - classe Euler del Pezzo deg 2']}
  • Theorem 1.4: Proposition \ref{['THO - classe Euler del Pezzo degre 4']} below
  • Theorem 1.5: Theorem \ref{['THO - classe Euler non orientable']} below
  • Proposition 2.1.1: Ayoub_6foncteurs
  • Theorem 2.1.2: Six functors on $\operatorname{S}\!\mathcal{H}^{}\!\left(S\right)$, Ayoub_6foncteurs
  • Definition 2.1.3: Suspension by a locally free sheaf
  • Proposition 2.1.4: Thom space of a locally free sheaf
  • Remark 2.1.5
  • Remark 2.1.6
  • ...and 107 more