Resolutions of Collinearity Among Four Points in the Complex Projective Plane

• Main Author: Piercey, Victor Ian
• Other Authors: Hu, Yi
• Language: en
• Published: The University of Arizona. 2012
• Subjects: Mathematics
• Online Access: http://hdl.handle.net/10150/222831

By a theorem of Mnëv, every possible algebraic singularity occurs in certain incidence subspaces of (P²)ⁿ x (P²)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 Xᵒ(n) ⊂ (P²)n x (P²)^(n/2) along with its closure X(n), which is singular for n ≥ 3. The open strata Xᵒ(n) parameterizes configurations of n points in P² that are in general linear position. When n = 3, a smooth compactification of Xn 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 θᵢ (Theorem 2.12). Some of the possible blowups that would desingularize X₃ fit into the framework of the Atiyah flop. There are forgetful morphisms Fᵢ : X₄ → X₃ 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 (Proposition 4.33). 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 (Proposition 4.37). These generators and relations are used to prove that blowing up X₄ at F(j)⁻¹(ε); F(k)⁻¹(ε); F(l)⁻¹(ε); and Fᵢ⁻¹(τ), where ε and τ are strata in X₃, results in a smooth space (Theorem 5.24) whose boundary consists of smooth divisorial components (Theorem 5.29).