Weak Brill-Noether on Abelian Surfaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Coskun, Izzet Nuer, Howard Yoshioka, Kota |
| author_facet | Coskun, Izzet Nuer, Howard Yoshioka, Kota |
| contents | We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and let $\mathbf{v}=(r,ξ,a)$ be a Mukai vector on $X$ with $\mathbf{v}^2\ge 0$,$r>0$, and $ξ\cdot H>0$. We show that if $ρ(X)=1$ or $ρ(X)=2$ and $X$ contains an elliptic curve, then all the moduli spaces $M_{X,H}(\mathbf{v})$ satisfy weak Brill-Noether. Conversely, if $ρ(X)>2$ or $ρ(X)=2$ and $X$ does not contain an elliptic curve, we show that there are infinitely many moduli spaces $M_{X,H}(\mathbf{v})$ that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak Brill-Noether on Abelian Surfaces Coskun, Izzet Nuer, Howard Yoshioka, Kota Algebraic Geometry Primary 14D20, 14K99. Secondary: 14F06, 14F08 We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and let $\mathbf{v}=(r,ξ,a)$ be a Mukai vector on $X$ with $\mathbf{v}^2\ge 0$,$r>0$, and $ξ\cdot H>0$. We show that if $ρ(X)=1$ or $ρ(X)=2$ and $X$ contains an elliptic curve, then all the moduli spaces $M_{X,H}(\mathbf{v})$ satisfy weak Brill-Noether. Conversely, if $ρ(X)>2$ or $ρ(X)=2$ and $X$ does not contain an elliptic curve, we show that there are infinitely many moduli spaces $M_{X,H}(\mathbf{v})$ that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces. |
| title | Weak Brill-Noether on Abelian Surfaces |
| topic | Algebraic Geometry Primary 14D20, 14K99. Secondary: 14F06, 14F08 |
| url | https://arxiv.org/abs/2408.06095 |