Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class

E. Javier Elizondo; Alex Fink; Cristhian Garay López

doi:10.1016/j.jpaa.2025.107930

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arXiv · 2112.15334

Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class

E. Javier Elizondo·Alex Fink·Cristhian Garay López

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Abstract

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 $\lambda$. We give a complete answer for the case where $\lambda$ 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 $\mathbb{G}(2,5)$.

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E. Javier Elizondo, Alex Fink, Cristhian Garay López. 2021-12-31. Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class. https://doi.org/10.1016/j.jpaa.2025.107930

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