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arXiv · 2604.19253

The reciprocal complement of a surface

Dario Spirito
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Abstract

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 $\mathcal{R}(K[X,Y])$.

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Dario Spirito. 2026-04-21. The reciprocal complement of a surface. https://arxiv.org/abs/2604.19253

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