Geometry of Chow quotients of Grassmannians

Abstract

We consider Kapranov's Chow quotient compactification of the moduli space of ordered $n$-tuples of hyperplanes in $P^{r-1}$ in linear general position. For $r=2$, this is canonically identified with the Grothendieck-Knudsen compactification of $M_{0,n}$ which has, among others, the following nice properties:

(1) modular meaning: stable pointed rational curves;

(2) canonical description of limits of one-parameter degenerations;

(3) natural Mori theoretic meaning: log-canonical compactification.

We generalize (1) and (2) to all $(r,n)$, but we show that (3), which we view as the deepest, fails except possibly in the cases $(2,n)$, $(3,6)$, $(3,7)$, $(3,8)$, where we conjecture that it holds