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Motivic cycles on K3 double covers of del Pezzo surfaces

Ramesh Sreekantan

Abstract

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on the del Pezzo and is a generalization of a recent pre-print of Ken Sato arXiv: 2408.09102 where he considers the case of fourfold covers of ${\mathbb P}^2$ branched at a quartic curve.

Motivic cycles on K3 double covers of del Pezzo surfaces

Abstract

We construct motivic cohomology cycles in the group where is a K3 surface obtained as a double cover of a del Pezzo surface branched at a curve in . The construction uses (-1) curves on the del Pezzo and is a generalization of a recent pre-print of Ken Sato arXiv: 2408.09102 where he considers the case of fourfold covers of branched at a quartic curve.

Paper Structure

This paper contains 15 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. del Pezzo surfaces
  3. $(-1)$-curves on a del Pezzo surface
  4. K3 double covers of del Pezzo surfaces
  5. Motivic cycles
  6. Presentation of cycles in $H^3_{{\mathcal{M}}}(X,{\mathds Q}(2))$.
  7. The localization sequence
  8. Motivic cycles from $(-1)$-curves
  9. Construction
  10. Indecomposability
  11. Degree $d$ del Pezzo surfaces and the degeneracy loci
  12. Degree 1
  13. Degree 2
  14. Degree 3
  15. Rational curves on del Pezzo surfaces

Key Result

Theorem 2.1

Let $X=X_d$ be a smooth del Pezzo surface of degree $d$. Then Here general position means no three points lie on a line, no six points lie on a conic and no 8 points lie on a nodal cubic with one of them being the node.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 4 more