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

TL;DR

The paper introduces a new smooth moduli space for point configurations in the projective plane, based on a modified Chow quotient inspired by Kapranov's construction, improving smoothness for six points.

Contribution

It presents a novel version of Kapranov's Chow quotient that yields a smooth moduli space for six points in the projective plane.

Findings

The new construction produces a smooth moduli space for six points.

Both constructions serve as compactifications of generic point configurations.

The new method incorporates additional lines connecting point pairs, inspired by blow-up techniques.

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.