Abstract

We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c_2.

Highlights

  • It is well-known that a general curve of genus g ≤ 9 or g = 11 can be realized as a linear section of a primitively polarized K 3 surface, cf. [26,28]

  • The Lazarsfeld–Mukai bundle associated to a pencil on a smooth curve on the K 3 surface induced by an elliptic pencil on the surface is necessarily not stable, cf

  • Section 3: We prove that a general curve C of genus 4 is a linear section of a smooth K 3 surface S such that its two g31s are induced by two elliptic pencils |E1| and |E2| on S satisfying C ∼ E1 + E2, cf

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Summary

Introduction

It is well-known that a general curve of genus g ≤ 9 or g = 11 can be realized as a linear section of a primitively polarized K 3 surface, cf [26,28]. Using Mukai’s results, we will study projective models of Brill–Noether general K 3 surfaces of genus g ∈ {4, 6, 8} containing the maximal possible number of elliptic pencils of degree g 2. Section 3: We prove that a general curve C of genus 4 is a linear section of a smooth K 3 surface S such that its two g31s (which are well-known to be auto-residual) are induced by two elliptic pencils |E1| and |E2| on S satisfying C ∼ E1 + E2, cf Proposition 3.4.

Lattice polarized K 3 surfaces and their moduli spaces
K 3 surfaces of genus 4
K3 surfaces of genus 4 with an elliptic pencil of degree 3
K3 surfaces of genus 6
Lazarsfeld–Mukai bundles and their stability
K3 surfaces of genus 8
Dual Grassmannian and Schubert varieties
Maximal number of distinct elliptic pencils
A unirational construction of K3 surfaces with nine distinct elliptic pencils
The moduli map
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