The cohomology of Hyperquot schemes on curves via shifted Yangians in type A
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912971148492800 |
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| author | Kaushik, Archi |
| author_facet | Kaushik, Archi |
| contents | Let $V$ be a vector bundle of rank $r$ on a smooth projective complex curve $C$. The Hyperquot scheme $\text{F}^{n}\text{Quot}\,(V)$ is the moduli space of length $n$ flags of rank $r$ sub-sheaves of $V$. This article has two main results: First, we show that a certain shifted Yangian of $\mathfrak{sl}_{n+1}$ acts on $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$ by correspondences. Then, we define a family of $rn$ commuting Yangian operators which yields a natural basis for $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$. This generalises the work arXiv:2307.13671 of Marian and Negut, who proved the above results in the case $n=1$. The new feature, which makes this generalisation possible, is the use of so called skew-nested Quot schemes. The rank $1$ versions of these spaces, skew-nested Hilbert schemes, have been recently introduced by Sergej Monavari in the context of refined DT theory of local curves arXiv:2506.14359. In the present article, skew-nested Quot schemes appear as correspondences associated with iterated commutators of Yangian elements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16691 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The cohomology of Hyperquot schemes on curves via shifted Yangians in type A Kaushik, Archi Algebraic Geometry Representation Theory Let $V$ be a vector bundle of rank $r$ on a smooth projective complex curve $C$. The Hyperquot scheme $\text{F}^{n}\text{Quot}\,(V)$ is the moduli space of length $n$ flags of rank $r$ sub-sheaves of $V$. This article has two main results: First, we show that a certain shifted Yangian of $\mathfrak{sl}_{n+1}$ acts on $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$ by correspondences. Then, we define a family of $rn$ commuting Yangian operators which yields a natural basis for $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$. This generalises the work arXiv:2307.13671 of Marian and Negut, who proved the above results in the case $n=1$. The new feature, which makes this generalisation possible, is the use of so called skew-nested Quot schemes. The rank $1$ versions of these spaces, skew-nested Hilbert schemes, have been recently introduced by Sergej Monavari in the context of refined DT theory of local curves arXiv:2506.14359. In the present article, skew-nested Quot schemes appear as correspondences associated with iterated commutators of Yangian elements. |
| title | The cohomology of Hyperquot schemes on curves via shifted Yangians in type A |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2603.16691 |