Non-commutative quadric surfaces

TL;DR

This paper explores the geometry of non-commutative quadric surfaces derived from the Sklyanin algebra, revealing their singularities, Picard groups, and intersection theory, and introduces non-commutative analogues of classical geometric concepts.

Contribution

It introduces a detailed study of non-commutative quadric surfaces, including their singularities, rulings, Picard groups, and intersection pairing, extending classical geometry to non-commutative settings.

Findings

Identified four singular quadrics in the non-commutative pencil.

Characterized rulings by non-commutative lines and Picard groups.

Discovered that smooth non-commutative quadrics can contain a curve with self-intersection -2.

Abstract

The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of non-commutative quadric surfaces, We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by non-commutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a "curve" having self-intersection number -2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen-Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of non-commutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional non-commutative quadric hypersurfaces will apply to higher dimensional non-commutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.