Rational Complexity-One $\boldsymbol{T}$-Varieties Are Well-Poised
| Autor: | Nathan Ilten, Christopher Manon |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | International Mathematics Research Notices. 2019:4198-4232 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnx254 |
| Popis: | Given an affine rational complexity-one $T$-variety $X$, we construct an explicit embedding of $X$ in affine space ${\mathbb{A}}^n$. We show that this embedding is well-poised, that is, every initial ideal of $I_X$ is a prime ideal, and we determine the tropicalization ${\mathrm{Trop}}(X^\circ)$. We then study valuations of the coordinate ring $R_X$ of $X$ which respect the torus action, showing that for full rank valuations, the natural generators of $R_X$ form a Khovanskii basis. This allows us to determine Newton–Okounkov bodies of rational projective complexity-one $T$-varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibres of ${\mathbb{K}}^*\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a result of Süß and Ilten. |
| Databáze: | OpenAIRE |
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