Secant varieties of generalised Grassmannians
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915104950321152 |
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| author | Galgano, Vincenzo |
| author_facet | Galgano, Vincenzo |
| contents | Secant varieties of a homogeneously embedded generalised Grassmannian $G/P$ inherit the natural group action, and one can reduce the study of their local geometric properties to $G$-orbit representatives. The case of secant varieties of lines is particularly elegant as their $G$-orbits are induced by $P$-orbits in both $G/P$ and $\mathfrak{g}/\mathfrak{p}$. Parabolic orbits are a classical problem in Representation Theory, well understood when $G/P$ is cominuscule. Exploiting them, we provide a complete and uniform description of both the identifiable and singular loci of the secant variety of lines to any cominuscule variety. We also introduce a finer version of the $2$-nd Terracini locus, called $2$-nd strong-Terracini locus, and we determine it for cominuscule varieties. Finally, we analyse the non-cominuscule case of isotropic Grassmannians for comparison, and we highlight a few differences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Secant varieties of generalised Grassmannians Galgano, Vincenzo Algebraic Geometry Representation Theory 14L30, 14M15, 14N07 Secant varieties of a homogeneously embedded generalised Grassmannian $G/P$ inherit the natural group action, and one can reduce the study of their local geometric properties to $G$-orbit representatives. The case of secant varieties of lines is particularly elegant as their $G$-orbits are induced by $P$-orbits in both $G/P$ and $\mathfrak{g}/\mathfrak{p}$. Parabolic orbits are a classical problem in Representation Theory, well understood when $G/P$ is cominuscule. Exploiting them, we provide a complete and uniform description of both the identifiable and singular loci of the secant variety of lines to any cominuscule variety. We also introduce a finer version of the $2$-nd Terracini locus, called $2$-nd strong-Terracini locus, and we determine it for cominuscule varieties. Finally, we analyse the non-cominuscule case of isotropic Grassmannians for comparison, and we highlight a few differences. |
| title | Secant varieties of generalised Grassmannians |
| topic | Algebraic Geometry Representation Theory 14L30, 14M15, 14N07 |
| url | https://arxiv.org/abs/2410.08158 |