| By a theorem of Mnëv, every possible algebraic singularity occurs in certain incidence subspaces of (P2) n × [(P*)2]m for some n, m. These incidence subspaces are defined by conditions that include (1) certain points must lie on certain lines, (2) i of the n points coincide, and (3) j of the m lines coincide. To capture these incidence spaces, we define a configuration space X n° in (P2)n × [(P*)2]nC2 along with its closure Xn, which is singular for n > 2. The open strata X n° parameterizes configurations of n points in P2 that are in general linear position. When n = 3, a smooth compactification of X n was discovered by Schubert and refined by Semple in 1954. In order to desingularize X3, we add the net of conics through the three points. This is equivalent to blowing up at one of three strata &thetas;i. Some of the possible blowups that would desingularize X3 fit into the framework of the Atiyah flop. There are forgetful morphisms F i : X4 → X 3 that omit the ith point and the three lines incident on the ith point. The space X4 is homogeneously covered by open affines. Each open affine has a coordinate ring generated by coordinates pulled back from a flag variety and edge coordinates indexed by the edges of a graph Γ. The relations are given by certain linear relations, quadratic relations that come from the triangular subconfigurations, and cubic relations indexed by hexagons in the graph Γ. These generators and relations are used to prove that blowing up X4 at Fj-1(ε), Fk-1(ε), Fl-1(ε), and Fi-1(τ), where ε and τ are strata in X3, results in a smooth space whose boundary consists of smooth divisorial components. |
