Projective models for Hilbert squares of $K3$ surfaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909821436952576 |
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| author | Ortiz, Ángel David Ríos Rojas, Andrés Song, Jieao |
| author_facet | Ortiz, Ángel David Ríos Rojas, Andrés Song, Jieao |
| contents | For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective models for Hilbert squares of $K3$ surfaces Ortiz, Ángel David Ríos Rojas, Andrés Song, Jieao Algebraic Geometry 14J28, 14J42, 14M12 For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry. |
| title | Projective models for Hilbert squares of $K3$ surfaces |
| topic | Algebraic Geometry 14J28, 14J42, 14M12 |
| url | https://arxiv.org/abs/2510.02065 |