More birational involutions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914033257414656 |
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| author | Beri, Pietro Manivel, Laurent |
| author_facet | Beri, Pietro Manivel, Laurent |
| contents | For $S$ a very general polarized K3 surface of degree $8n-6$, we describe in geometrical terms a birational involution of the Hilbert scheme $S^{[n]}$ of $n$ points on the surface, whose existence was established from lattice theoretical considerations. In a previous work we studied this involution for $n=3$, with the help of the exceptional Lie group $G_2$, since the Mukai model of $S$ is embedded in its projectivized Lie algebra. Here we use different, more general arguments to show that some important features of the birational involution persist for $n\ge 4$. In particular, we describe the indeterminacy locus of the involution in terms of a Mori contraction, and deduce that it is birational to a $\mathbb{P}^2$-fibration over a moduli space of sheaves on $S$, that also admits a degree two nef and big line bundle and an induced birational involution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | More birational involutions Beri, Pietro Manivel, Laurent Algebraic Geometry 14J60, 14J28, 14E07, 14F08, 14J42 For $S$ a very general polarized K3 surface of degree $8n-6$, we describe in geometrical terms a birational involution of the Hilbert scheme $S^{[n]}$ of $n$ points on the surface, whose existence was established from lattice theoretical considerations. In a previous work we studied this involution for $n=3$, with the help of the exceptional Lie group $G_2$, since the Mukai model of $S$ is embedded in its projectivized Lie algebra. Here we use different, more general arguments to show that some important features of the birational involution persist for $n\ge 4$. In particular, we describe the indeterminacy locus of the involution in terms of a Mori contraction, and deduce that it is birational to a $\mathbb{P}^2$-fibration over a moduli space of sheaves on $S$, that also admits a degree two nef and big line bundle and an induced birational involution. |
| title | More birational involutions |
| topic | Algebraic Geometry 14J60, 14J28, 14E07, 14F08, 14J42 |
| url | https://arxiv.org/abs/2509.10130 |