Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$
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| Format: | Preprint |
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2024
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| _version_ | 1866913196703481856 |
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| author | Biswas, Indranil Garge, Shripad M. Hanumanthu, Krishna |
| author_facet | Biswas, Indranil Garge, Shripad M. Hanumanthu, Krishna |
| contents | Let $X$ be an irreducible smooth projective curve defined over $\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When $r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07478 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$ Biswas, Indranil Garge, Shripad M. Hanumanthu, Krishna Algebraic Geometry 14H60, 14G15, 11G25 Let $X$ be an irreducible smooth projective curve defined over $\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When $r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki. |
| title | Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$ |
| topic | Algebraic Geometry 14H60, 14G15, 11G25 |
| url | https://arxiv.org/abs/2401.07478 |