Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sections of K3 surfaces with Picard number two and Mercat's conjecture

Marian Aprodu, Laura Filimon

Abstract

Farkas and Ortega found counterexamples to Mercat's conjecture by restricting to a hyperplane section $C$ some suitable rank-two vector bundles on a $K3$ surface whose Picard group is generated by $C$ and another very ample divisor. We prove that the same bundles produce other counterexamples by restriction to hypersurface sections $C_n\in|nC|$ for all $n\ge 2$. In the process, we compute the Clifford indices of the corresponding hypersurface sections $C_n$, noting their non-generic nature for $n\ge 2$. A key ingredient to prove the (semi)stability of the restricted bundles, is Green's Explicit $H^0$ Lemma. In what concerns the (semi)stability, although general restriction theorems as demonstrated by Flenner or Feyzbakhsh are applicable for sufficiently large, explicit values of $n$, our approach works for all $n\ge 2$. It is also worth noting that our proof deviates slightly from the one of Farkas-Ortega. Employing the same strategy leads to an enhancement of the main result of a paper of Sengupta.

Sections of K3 surfaces with Picard number two and Mercat's conjecture

Abstract

Farkas and Ortega found counterexamples to Mercat's conjecture by restricting to a hyperplane section some suitable rank-two vector bundles on a surface whose Picard group is generated by and another very ample divisor. We prove that the same bundles produce other counterexamples by restriction to hypersurface sections for all . In the process, we compute the Clifford indices of the corresponding hypersurface sections , noting their non-generic nature for . A key ingredient to prove the (semi)stability of the restricted bundles, is Green's Explicit Lemma. In what concerns the (semi)stability, although general restriction theorems as demonstrated by Flenner or Feyzbakhsh are applicable for sufficiently large, explicit values of , our approach works for all . It is also worth noting that our proof deviates slightly from the one of Farkas-Ortega. Employing the same strategy leads to an enhancement of the main result of a paper of Sengupta.
Paper Structure (4 sections, 4 theorems, 33 equations)

This paper contains 4 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Basic properties of Lazarsfeld--Mukai bundles
  3. Clifford indices of hypersurface sections of a $K3$ surface with Picard number two
  4. New counterexamples to Mercat's conjecture

Key Result

Lemma 3.1

Put $E=C-D$. Then $E^2=0$, $h^0(S,\mathcal{O}_S(E)) = 2$ and $h^1(S,\mathcal{O}_S(E))=0$.

Theorems & Definitions (6)

  • Lemma 3.1
  • Theorem 3.2
  • Remark
  • Theorem 4.1
  • Remark
  • Theorem 4.2