Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.08389 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929210839269376 |
|---|---|
| author | Aprodu, Marian Filimon, Laura |
| author_facet | Aprodu, Marian Filimon, Laura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sections of K3 surfaces with Picard number two and Mercat's conjecture Aprodu, Marian Filimon, Laura Algebraic Geometry 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. |
| title | Sections of K3 surfaces with Picard number two and Mercat's conjecture |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2401.08389 |