A version of Kapranov's Chow quotient and smooth moduli space of point configurations in $\PP^2$

Abstract

A new moduli space for configurations of $n$ ordered points in a projective plane, which is a version of Kapranov's "Chow quotient of Grassmanians" is introduced. The new construction is a Chow quotient as well but with additional lines connecting pairs of marked points (inspired by the idea of blow up). Both Kapranov's construction and the new construction provide an algebraic variety which is a compactification for the space of generic configurations of $n$ distinct points in projective plane. The difference is, that in Kapranov's construction, the space for configurations of 6 points in two-dimensional plane is not smooth; while with the new construction, the space for configuration of 6 points in plane is smooth.