Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs
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| Format: | Preprint |
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2026
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| _version_ | 1866914501093228544 |
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| author | Gangopadhyay, Chandranandan Ito, Atsushi |
| author_facet | Gangopadhyay, Chandranandan Ito, Atsushi |
| contents | Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber $\mathcal Q_L$ over ${\rm Pic}^n(C)$ for $n \gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb Q$-factorial modifications (SQMs) of ${\rm Quot}_C(E^0, k, n)$ and $\mathcal Q_L$. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that $\mathcal Q_L$ is a Mori dream space and that the determinant morphism ${\rm Quot}_C(E^0, k, n) \to {\rm Pic}^n(C)$ is a Mori dream morphism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21303 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs Gangopadhyay, Chandranandan Ito, Atsushi Algebraic Geometry 14J60 Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber $\mathcal Q_L$ over ${\rm Pic}^n(C)$ for $n \gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb Q$-factorial modifications (SQMs) of ${\rm Quot}_C(E^0, k, n)$ and $\mathcal Q_L$. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that $\mathcal Q_L$ is a Mori dream space and that the determinant morphism ${\rm Quot}_C(E^0, k, n) \to {\rm Pic}^n(C)$ is a Mori dream morphism. |
| title | Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs |
| topic | Algebraic Geometry 14J60 |
| url | https://arxiv.org/abs/2604.21303 |