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 $C$ 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.