Chow quotients of Grassmannian I

TL;DR

This paper introduces a new compactification of the space of projective configurations using Chow varieties, differing from GIT quotients, and relates it to moduli spaces of stable curves and blow-ups of projective spaces.

Contribution

It presents a novel compactification of configuration spaces via Chow quotients, connecting orbit closures to moduli spaces and Veronese varieties, and offers a new perspective on these geometric objects.

Findings

Constructs a compactification different from GIT quotients.

Relates the compactification to moduli space of stable curves.

Describes the orbit closures as Veronese varieties.

Abstract

We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This compactification differs considerably from Mumford's geometric invariant theory quotient. It is obtained by considering limit position (in the Chow variety) of the closures of generic orbits. The same result will be obtained if we study orbits of the maximal torus on the Grassmannian $G(k,n)$. We study in detail the closures of the torus orbits and their "visible contours" which are Veronese varieties in the Grassmannian. For points on $P^1$ our construction gives the Grothemdieck - Knudsen moduli space of stable $n$ -punctured curves of genus 0. The "Chow quotient" interpretation of this space permits us to represent it as a blow up of a projective space.