The reciprocal complement of a surface
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908983321690112 |
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| author | Spirito, Dario |
| author_facet | Spirito, Dario |
| contents | We study the reciprocal complement $\mathcal{R}(D)$ of a two-dimensional finitely generated $K$-algebra $D$ by linking it with the properties of a surface with coordinate ring $D$. We give several sufficient criteria to have $\dim\mathcal{R}(D)=2$, and we use them to show several explicit examples; in particular, we determine the dimension of $\mathcal{R}(D)$ when $D$ is the quotient of $K[X,Y,Z]$ by an irreducible polynomial of degree $2$. We also study the integral closure of the localizations of $\mathcal{R}(K[X,Y])$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The reciprocal complement of a surface Spirito, Dario Commutative Algebra Algebraic Geometry We study the reciprocal complement $\mathcal{R}(D)$ of a two-dimensional finitely generated $K$-algebra $D$ by linking it with the properties of a surface with coordinate ring $D$. We give several sufficient criteria to have $\dim\mathcal{R}(D)=2$, and we use them to show several explicit examples; in particular, we determine the dimension of $\mathcal{R}(D)$ when $D$ is the quotient of $K[X,Y,Z]$ by an irreducible polynomial of degree $2$. We also study the integral closure of the localizations of $\mathcal{R}(K[X,Y])$. |
| title | The reciprocal complement of a surface |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2604.19253 |