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On strictly elliptic K3 surfaces and del Pezzo surfaces

Paola Comparin, Pedro Montero, Yulieth Prieto-Montañez, Sergio Troncoso

Abstract

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo surfaces with log-terminal singularities of Gorenstein index two and K3 surfaces with non-symplectic involutions of elliptic type: the latter surfaces are realized as appropriate double covers obtained from the former ones. The main application of this correspondence is in the study of linear systems that induce elliptic fibrations on K3 surfaces admitting a strictly elliptic non-symplectic involution, i.e., whose fixed locus consists of a single curve of genus $g\geq 2$. The obtained results are similar to those achieved by Garbagnati and Salgado for jacobian elliptic fibrations.

On strictly elliptic K3 surfaces and del Pezzo surfaces

Abstract

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo surfaces with log-terminal singularities of Gorenstein index two and K3 surfaces with non-symplectic involutions of elliptic type: the latter surfaces are realized as appropriate double covers obtained from the former ones. The main application of this correspondence is in the study of linear systems that induce elliptic fibrations on K3 surfaces admitting a strictly elliptic non-symplectic involution, i.e., whose fixed locus consists of a single curve of genus . The obtained results are similar to those achieved by Garbagnati and Salgado for jacobian elliptic fibrations.
Paper Structure (7 sections, 11 theorems, 13 equations, 3 figures, 3 tables)

This paper contains 7 sections, 11 theorems, 13 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Conic bundles and smooth del Pezzo surfaces
  4. Non-symplectic involutions on K3 surfaces
  5. Strictly elliptic involutions
  6. Examples and Néron-Severi lattices
  7. Singular fibers and bisections

Key Result

Theorem 2.1

Let $Z$ be a smooth del Pezzo surface of degree $d$. Then, $1\leq d \leq 9$ and we have that: Here, we say that the points are in general position if the following hold: Conversely, any blow-up of $r\leq 8$ points in general position is a smooth del Pezzo surface.

Figures (3)

  • Figure 1: All possible invariants $(r,a,\delta)$.
  • Figure 2: Strictly elliptic K3 surfaces and smooth del Pezzo surfaces as in Proposition \ref{['prop: K3 of strictly elliptic type']} (2).
  • Figure 3: Elliptic fibrations on strictly elliptic K3 surfaces and conic bundles on del Pezzo surfaces.

Theorems & Definitions (38)

  • Theorem 2.1
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 28 more