A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface

TL;DR

This paper constructs a non-commutative homogeneous coordinate ring for a degree six del Pezzo surface, linking algebraic and geometric properties and establishing its noetherian and homological characteristics.

Contribution

It introduces a new non-commutative coordinate ring for the del Pezzo surface and proves its geometric and algebraic properties, including being a twisted homogeneous coordinate ring.

Findings

R is a twisted homogeneous coordinate ring for B_3

R is a noetherian domain of global dimension three

The generic simple R-module has dimension six

Abstract

Let R be the free algebra on x and y modulo the relations x^5=yxy and y^2=xyx endowed with the grading deg x=1 and deg y=2. Let B_3 denote the blow up of the projective plane at three non-colliear points. The main result in this paper is that the category of quasi-coherent sheaves on B_3 is equivalent to the quotient of the category of graded R-modules modulo the full subcategory of modules M such that for each m in M, $(x,y)^nm=0$ for n sufficiently large. This is proved by showing the R is a twisted homogeneous coordinate ring (in the sense of Artin and Van den Bergh) for B_3. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B_3. For example, the generic simple R-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, and Van den Bergh, imply that R is a noetherian domain of global dimension three, and has other good homological properties.