On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911900358410240 |
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| author | Ciliberto, Ciro Verra, Alessandro |
| author_facet | Ciliberto, Ciro Verra, Alessandro |
| contents | Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0--dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree $152$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$ Ciliberto, Ciro Verra, Alessandro Algebraic Geometry Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0--dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree $152$. |
| title | On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2406.00822 |