Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911446871310336 |
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| author | Knutsen, Andreas Leopold Torelli, Sara |
| author_facet | Knutsen, Andreas Leopold Torelli, Sara |
| contents | Under the assumption that the adjusted Brill-Noether number $\widetildeρ$ is at least $-g$, we prove that the Brill-Noether loci in $\mathcal{M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to $\mathcal{M}_g$ is dominant if $n+\widetildeρ \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski.
In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with non-general behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces Knutsen, Andreas Leopold Torelli, Sara Algebraic Geometry 14C25, 14H10, 14H51, 14J28 Under the assumption that the adjusted Brill-Noether number $\widetildeρ$ is at least $-g$, we prove that the Brill-Noether loci in $\mathcal{M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to $\mathcal{M}_g$ is dominant if $n+\widetildeρ \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski. In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with non-general behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves. |
| title | Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces |
| topic | Algebraic Geometry 14C25, 14H10, 14H51, 14J28 |
| url | https://arxiv.org/abs/2411.14562 |