The reciprocal complement of a surface

TL;DR

This paper investigates the properties and dimension of the reciprocal complement of a surface algebra, providing criteria, explicit examples, and studying related integral closures.

Contribution

It introduces new criteria for the dimension of the reciprocal complement and analyzes specific cases involving quotient rings and integral closures.

Findings

Determined the dimension of the reciprocal complement for certain quotient rings.

Provided explicit examples illustrating the criteria for dimension.

Studied the integral closure of localizations of the reciprocal complement.

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