Some noncommutative projective surfaces of GK-dimension 4

TL;DR

This paper constructs new noncommutative projective surfaces of GK-dimension 4 that are birational to P2, revealing novel properties and disproving existing conjectures in the field.

Contribution

It introduces a family of noncommutative surfaces with specific algebraic properties, challenging previous assumptions and extending the understanding of noncommutative algebraic geometry.

Findings

Constructed noncommutative surfaces of GK-dimension 4 birational to P2.

Disproved a conjecture regarding noetherian properties of such algebras.

Showed these algebras are Koszul with global dimension 4, but not Artin-Schelter Gorenstein.

Abstract

We construct a family of connected graded domains of GK-dimension 4 that are birational to P2, and show that the general member of this family is noetherian. This disproves a conjecture of the first author and Stafford. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin-Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander-Buchsbaum formula also fails to hold for our family. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on P1xP1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.