Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915816882044928 |
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| author | Postinghel, Elisa Prendergast-Smith, Artie |
| author_facet | Postinghel, Elisa Prendergast-Smith, Artie |
| contents | We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of $\mathbb P^n \times \mathbb P^{n+1}$ play a central role in the study of the birational geometry of $X^{n,n+1}_s$, its blowup in $s$ points in general position. We show that $X^{n,n+1}_s$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X^{3,4}_6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$ Postinghel, Elisa Prendergast-Smith, Artie Algebraic Geometry 14C20, 14J45, 14J70, 14N07 We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of $\mathbb P^n \times \mathbb P^{n+1}$ play a central role in the study of the birational geometry of $X^{n,n+1}_s$, its blowup in $s$ points in general position. We show that $X^{n,n+1}_s$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X^{3,4}_6$. |
| title | Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$ |
| topic | Algebraic Geometry 14C20, 14J45, 14J70, 14N07 |
| url | https://arxiv.org/abs/2412.19364 |