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Pseudo-automorphisms of rational threefolds and Kummer surfaces

Zhuang He

Abstract

Kummer surfaces are special quartic surfaces that admit $16$ nodes. The automorphisms of K3 Kummer surfaces are rich and complicated. Based on the results of Keum and Kondō, and as a continuation of the recent result by He and Yang, we lift $45$ classically known automorphisms of Kummer surfaces to pseudo-automorphisms of a threefold, the blow-up of $\mathbb{P}^3$ along $6$ points and $15$ lines. We give a description of a fundamental domain of the group generated by all the known pseudo-automorphisms on this threefold.

Pseudo-automorphisms of rational threefolds and Kummer surfaces

Abstract

Kummer surfaces are special quartic surfaces that admit nodes. The automorphisms of K3 Kummer surfaces are rich and complicated. Based on the results of Keum and Kondō, and as a continuation of the recent result by He and Yang, we lift classically known automorphisms of Kummer surfaces to pseudo-automorphisms of a threefold, the blow-up of along points and lines. We give a description of a fundamental domain of the group generated by all the known pseudo-automorphisms on this threefold.
Paper Structure (22 sections, 27 theorems, 70 equations, 2 tables)

This paper contains 22 sections, 27 theorems, 70 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Kummer surfaces
  3. Nodes and tropes
  4. Cones of divisors
  5. Classical involutions
  6. 16 translations $t_\alpha$
  7. The switch $\sigma$
  8. 16 projections $p_\alpha$
  9. 16 correlations $q_\alpha$
  10. Hutchinson-Göpel and Hutchinson-Weber involutions; Keum's automorphisms
  11. Göpel tetrads
  12. Weber hexads
  13. HG-type Pseudo-automorphisms
  14. The threefold and the anticanonical surface
  15. The complete linear system
  16. ...and 7 more sections

Key Result

Proposition 1.1

(Proposition injhom) The group homomorphism $u:PsAut(X)\to \mathop{\mathrm{Aut}}\nolimits(S)$ is injective.

Theorems & Definitions (31)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 21 more