Castelnuovo-Mumford regularity of toric varieties with at most one singular point
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| author | García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario |
| author_facet | García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario |
| contents | We establish upper bounds for the Castelnuovo--Mumford regularity of the coordinate ring of a simplicial projective toric variety with at most one singular point. In the smooth case, our results recover the bound of Herzog and Hibi [Proc. Amer. Math. Soc. 131 (2003), 2641--2647], and therefore the Eisenbud--Goto bound. Furthermore, when the variety has exactly one singular point and dimension at least $3$, we prove that its regularity also satisfies the Eisenbud--Goto bound. The proof combines combinatorial and homological methods: we study the asymptotic behavior of the sumsets associated to the toric variety and relate it to Castelnuovo--Mumford regularity via a Hochster-like formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18162 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Castelnuovo-Mumford regularity of toric varieties with at most one singular point García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario Commutative Algebra Algebraic Geometry Combinatorics Primary 13D02, Secondary 14Q20, 20M50, 11B13 We establish upper bounds for the Castelnuovo--Mumford regularity of the coordinate ring of a simplicial projective toric variety with at most one singular point. In the smooth case, our results recover the bound of Herzog and Hibi [Proc. Amer. Math. Soc. 131 (2003), 2641--2647], and therefore the Eisenbud--Goto bound. Furthermore, when the variety has exactly one singular point and dimension at least $3$, we prove that its regularity also satisfies the Eisenbud--Goto bound. The proof combines combinatorial and homological methods: we study the asymptotic behavior of the sumsets associated to the toric variety and relate it to Castelnuovo--Mumford regularity via a Hochster-like formula. |
| title | Castelnuovo-Mumford regularity of toric varieties with at most one singular point |
| topic | Commutative Algebra Algebraic Geometry Combinatorics Primary 13D02, Secondary 14Q20, 20M50, 11B13 |
| url | https://arxiv.org/abs/2603.18162 |