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

Let R be the free C-algebra on x and y modulo the relations x5=yxy and y2=xyx endowed with the Z-grading and . The ring R appears, in somewhat hidden guise, in a paper on quiver gauge theories. Let B3 denote the blow up of CP2 at three non-colinear points. The main result in this paper is that the category of quasi-coherent OB3-modules is equivalent to the quotient of the category of Z-graded R-modules modulo the full subcategory of modules that are the sum of their finite dimensional submodules. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B3. For example, the generic simple R-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, Van den Bergh, and Stephenson implies that R is a noetherian domain of global dimension three.