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Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

Alice Garbagnati

Abstract

Given a symplectic involution $ι$ on a K3 surface $X$, the desingularization $Y$ of $X/ι$ is still a K3 surface, which in general has a different Néron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the Néron--Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration.

Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

Abstract

Given a symplectic involution on a K3 surface , the desingularization of is still a K3 surface, which in general has a different Néron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the Néron--Severi groups of and are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy . Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that . There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which . The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration.
Paper Structure (27 sections, 32 theorems, 64 equations, 1 figure)

This paper contains 27 sections, 32 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. K3 surfaces with symplectic a involution and their specializations
  3. Geometric description of the case $d=1$
  4. The $(U\oplus N)$-polarized family and the van Geemen Sarti involution
  5. Preliminaries on $\mathcal{N}$
  6. Cohomological action of the van Geemen--Sarti involution
  7. Specializations of $X$ and $Y$ to K3 surfaces with Picard number 11
  8. The surfaces $X_{11}$
  9. The surface $Y_{11}$
  10. Some specializations of $X$ and $Y$ to K3 surfaces with Picard number 12
  11. Self maps of the transcendental lattice and specializations
  12. The main results and a specialization of $X$ and $Y$ to a K3 surface with Picard number 20
  13. Proof of Theorem \ref{['theorem: action selfmap']}
  14. Strategy of the proof
  15. Construction of the basis of $(T_X)_{\mathbb{Q}}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

There are infinitely many 12-dimensional families of projective K3 surfaces whose generic member, $X$, admits a symplectic involution $\iota$ and $NS(Y)\simeq NS(X)$, where $Y$ is the minimal resolution of $X/\iota$.

Figures (1)

  • Figure 1: Specializations: the 2-torsion section $T$ is red, and we assume that the trivial component of the $I_4$ and $I_8$ fibers is the horizontal lower component and the trivial component of the $I_2$-fibers is the left component.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Theorem 2.6
  • ...and 45 more