Singular loci of Schubert varieties and the Lookup Conjecture in type $\tilde A_{2}$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911971348054016 |
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| author | Boe, Brian D. Graham, William |
| author_facet | Boe, Brian D. Graham, William |
| contents | We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the results of Graham and Li for spiral elements, this allows us to explicitly identify the maximal singular and nrs points in any Schubert variety of type $\tilde{A}_2$. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$, where every Schubert variety is rationally smooth). As a consequence, we deduce that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through $x$ which are stable under the action of a maximal torus, as is true for Schubert varieties in (finite) type $A$. Combined with the work of Graham and Li for spiral Schubert varieties, this implies the Lookup Conjecture for $\tilde{A}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular loci of Schubert varieties and the Lookup Conjecture in type $\tilde A_{2}$ Boe, Brian D. Graham, William Algebraic Geometry Combinatorics Representation Theory 14M15 (Primary) 14B05, 05E14 (Secondary) We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the results of Graham and Li for spiral elements, this allows us to explicitly identify the maximal singular and nrs points in any Schubert variety of type $\tilde{A}_2$. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$, where every Schubert variety is rationally smooth). As a consequence, we deduce that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through $x$ which are stable under the action of a maximal torus, as is true for Schubert varieties in (finite) type $A$. Combined with the work of Graham and Li for spiral Schubert varieties, this implies the Lookup Conjecture for $\tilde{A}_2$. |
| title | Singular loci of Schubert varieties and the Lookup Conjecture in type $\tilde A_{2}$ |
| topic | Algebraic Geometry Combinatorics Representation Theory 14M15 (Primary) 14B05, 05E14 (Secondary) |
| url | https://arxiv.org/abs/2407.02338 |