Geometry of genus sixteen K3 surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915647452086272 |
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| author | Han, Frederic |
| author_facet | Han, Frederic |
| contents | Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in $\mathbb{P}_9$ with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai's unirationalization of the moduli space. This linear system gives a double cover of $\mathbb{P}_9$ ramified on a degree $10$ hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus $16$ is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of genus sixteen K3 surfaces Han, Frederic Algebraic Geometry 14J32, 14J35, 14M15, 14J70, 14J28 Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in $\mathbb{P}_9$ with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai's unirationalization of the moduli space. This linear system gives a double cover of $\mathbb{P}_9$ ramified on a degree $10$ hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus $16$ is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors. |
| title | Geometry of genus sixteen K3 surfaces |
| topic | Algebraic Geometry 14J32, 14J35, 14M15, 14J70, 14J28 |
| url | https://arxiv.org/abs/2501.16013 |