The reciprocal complement of a surface

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

• The paper presents a framework establishing criteria for when the reciprocal complement of a surface attains maximal Krull dimension.
• Reduction techniques via f-transforms connect surface localizations to curve dimension results, offering practical insights into the behavior at infinity.
• Explicit examples on quadrics and cubics illustrate how irreducibility, nonsingularity, and nonconstant units in D affect integral closure and Noetherian properties.

The Reciprocal Complement of a Surface

Introduction and Motivation

This paper addresses the algebraic and geometric structure of the reciprocal complement $R(D)$ for integral domains $D$ associated with algebraic surfaces, extending previous geometric characterizations from the 1-dimensional (curve) case to surfaces. For a two-dimensional, finitely generated $K$-algebra $D$, $R(D)$ is defined as the subring of the quotient field $Q(D)$ generated by the inverses of the nonzero elements of $D$. Understanding the Krull dimension of $R(D)$, and the conditions under which it achieves the maximal value ($2$ for surfaces), links this algebraic construction to explicit properties of the underlying algebraic variety. The work also examines local and integral closure properties of $R(D)$, with an emphasis on localizations and cases such as quadrics.

Main Constructions and Core Results

The author systematically generalizes prior results for curves, particularly dropping the restriction that the base field $D$0 is algebraically closed. When $D$1 is a coordinate ring of a curve (transcendence degree one), the dimension and field structure of $D$2 are controlled by the behavior at infinity on the projective closure $D$3 of the affine curve. Specifically, $D$4 is a field if and only if the set of points at infinity is not a singleton, i.e., iff $D$5.

The transition to surfaces (transcendence degree two) introduces additional complexity, as two-dimensional varieties can admit multiple non-equivalent embeddings with non-isomorphic behavior at infinity. The author does not provide a complete classification for the possible dimensions of $D$6 in this setting, but presents several sufficient criteria for maximality --- that is, for $D$7. These criteria rely crucially on geometric arguments about the projective closure, the structure of the set $D$8 at infinity, and the existence of suitable regular functions.

Reduction Techniques and Localization

A pivotal technical component is a reduction technique leveraging the structure of principal open sets and the introduction of the $D$9-transform: Given a regular function $K$0 on a surface $K$1, the paper considers the variety in $K$2 (for a new transcendental $K$3) defined by the equations of $K$4 together with $K$5. The coordinate ring of this $K$6-transform, over $K$7, allows the application of dimension results for curves to the analysis of surface localizations, thereby bridging the gap between the geometric and algebraic perspectives.

Sufficient Conditions and Examples

The paper catalogs several explicit sufficient conditions for $K$8. These include:

• If the projective closure at infinity is irreducible and contains nonsingular points, then the reciprocal complement is nontrivial and its dimension can reach 2, provided there exists a function intersecting infinity in a unique nonsingular point.
• For coordinate rings of quadrics (quotients of $K$9 by an irreducible degree-2 polynomial), $D$0 attains dimension 2 unless the quadric is rational, in which case the dimension drops and $D$1 acquires nonconstant units.
• For certain classes of cubics and higher-degree surfaces, similar geometric criteria (irreducibility, existence of inflection points, or tangency conditions at infinity) ensure the maximal Krull dimension.

Moreover, the author investigates the effect of the arithmetic of the base field, showing that the non-closure of $D$2 can, somewhat counterintuitively, make it easier to guarantee that $D$3 has maximal dimension by forcing points at infinity to merge under field automorphisms.

Integral Closure and Noetherian Properties

A notable technical result is the demonstration that $D$4, even for the polynomial ring in two variables, is neither Noetherian nor integrally closed. The author gives a geometric proof (emphasizing open and local behavior) that supplements computational approaches found in earlier work. Moreover, the localizations of $D$5 can fail to be DVRs, and the presence of nonconstant units in $D$6 precisely corresponds to dimension drops in $D$7.

Implications and Theoretical Significance

The work advances the understanding of reciprocal complement constructions in commutative algebra and algebraic geometry, emphasizing the delicate interaction between algebraic properties of $D$8 and geometric features of the underlying varieties. The results clarify the connections between the appearance of nonconstant units, rationality, and the dimension of the reciprocal complement, yielding practical geometric tools for explicit computation in low-dimensional cases.

From a theoretical standpoint, these characterizations of $D$9 contribute to the structural understanding of non-Noetherian domains and their integral closures, offering new techniques for recognizing when a ring generated by reciprocals maintains "enough" dimension. The examples and criteria established are widely applicable to the study of algebraic surfaces, especially in the context of affine and projective embeddings, and their associated function rings.

Conclusion

The paper provides new geometric and algebraic criteria for the maximality of the Krull dimension of the reciprocal complement of surfaces, extending characterizations from curves and connecting the algebraic structure of $R(D)$0 to concrete properties at infinity of the associated varieties. Reduction techniques via $R(D)$1-transforms and careful analysis of units and localizations yield a partial but significant classification for surfaces, with explicit computations for quadrics, cubics, and specific classes of hypersurfaces. This work significantly enriches the toolkit for understanding the algebraic geometry of surfaces via their coordinate rings and their associated reciprocal complements (2604.19253).