Normality, factoriality and strong $F$-regularity of Lovász-Saks-Schrijver rings
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| Format: | Preprint |
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2024
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| _version_ | 1866916266896261120 |
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| author | Tolosa-Villarreal, Eliana |
| author_facet | Tolosa-Villarreal, Eliana |
| contents | Every simple finite graph $G$ has an associated Lovász-Saks-Schrijver ring $R_G(d)$ that is related to the $d$-dimensional orthogonal representations of $G$. The study of $R_G(d)$ lies at the intersection between algebraic geometry, commutative algebra and combinatorics. We find a link between algebraic properties such as normality, factoriality and strong $F$-regularity of $R_G(d)$ and combinatorial invariants of the graph $G$. In particular we prove that if $d \geq \text{pmd}(G)+k(G)$ then $R_G(d)$ is $F$-regular in finite characteristic and rational singularity in characteristic $0$ and furthermore if $d \geq \text{pmd}(G)+k(G)+1$ then $R_G(d)$ is UFD. Here $\text{pmd}(G)$ is the positive matching decomposition number of $G$ and $k(G)$ is its degeneracy number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normality, factoriality and strong $F$-regularity of Lovász-Saks-Schrijver rings Tolosa-Villarreal, Eliana Commutative Algebra Combinatorics Every simple finite graph $G$ has an associated Lovász-Saks-Schrijver ring $R_G(d)$ that is related to the $d$-dimensional orthogonal representations of $G$. The study of $R_G(d)$ lies at the intersection between algebraic geometry, commutative algebra and combinatorics. We find a link between algebraic properties such as normality, factoriality and strong $F$-regularity of $R_G(d)$ and combinatorial invariants of the graph $G$. In particular we prove that if $d \geq \text{pmd}(G)+k(G)$ then $R_G(d)$ is $F$-regular in finite characteristic and rational singularity in characteristic $0$ and furthermore if $d \geq \text{pmd}(G)+k(G)+1$ then $R_G(d)$ is UFD. Here $\text{pmd}(G)$ is the positive matching decomposition number of $G$ and $k(G)$ is its degeneracy number. |
| title | Normality, factoriality and strong $F$-regularity of Lovász-Saks-Schrijver rings |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2405.20480 |