Geometry of Chow Quotients of Grassmannians

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 M0,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.