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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.15334 |
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| _version_ | 1866916645380816896 |
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| author | Elizondo, E. Javier Fink, Alex López, Cristhian Garay |
| author_facet | Elizondo, E. Javier Fink, Alex López, Cristhian Garay |
| contents | Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and having given homology class $λ$. We give a complete answer for the case where $λ$ is the class of a $T$-orbit, and partial results for other cases, using techniques inspired by matroid theory. This problem has applications to the computation of the Euler-Chow series for Grassmannians of projective lines: we calculate the series for 3-cycles in $\mathbb{G}(2,4)$ and carry out partial calculations for $\mathbb{G}(2,5)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_15334 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class Elizondo, E. Javier Fink, Alex López, Cristhian Garay Algebraic Geometry Combinatorics Primary: 14C25, Secondary: 05B35 Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and having given homology class $λ$. We give a complete answer for the case where $λ$ is the class of a $T$-orbit, and partial results for other cases, using techniques inspired by matroid theory. This problem has applications to the computation of the Euler-Chow series for Grassmannians of projective lines: we calculate the series for 3-cycles in $\mathbb{G}(2,4)$ and carry out partial calculations for $\mathbb{G}(2,5)$. |
| title | Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class |
| topic | Algebraic Geometry Combinatorics Primary: 14C25, Secondary: 05B35 |
| url | https://arxiv.org/abs/2112.15334 |