  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.

Rings and Algebras;Algebraic Geometry