Geometry of Chow quotients of Grassmannians. (English) Zbl 1107.14026

Keel, Sean; Tevelev, Jenia

Duke Math. J. 134, No. 2, 259-311 (2006).

M. M. Kapranov [J. Algebr. Geom. 2, No. 2, 239–262 (1993; Zbl 0790.14020)] gave a compactification \(\bar X(r,n)\) of the moduli space \(X(r,n)\) of ordered \(r\)-tuples of hyperplanes in \({\mathbb P}^{r-1}\) in linear general position. The construction works by looking at the Chow points of orbit closures of an action of an algebraic group, in this case \(\text{ PGL}_r\), on a Grassmannian. If \(r=2\) one recovers the usual compactification \(\bar M_{0,n}\) of \(M_{0,n}\), the space of \(n\) ordered points on \({\mathbb P}^1\). Like \(\bar M_{0,n}\), \(\bar X(r,n)\) carries a family of pairs and the fibres have good (actually toroidal) singularities. All these facts were proved by Kapranov [loc. cit.] and L. Lafforgue [Invent. Math. 136, No. 1, 233–271 (1999; Zbl 0965.14024)].
It was observed by Hacking, and independently by the authors (the proof is not given here), that \(\bar X(r,n)\) admits a moduli interpretation as a moduli space of certain semi log canonical pairs, which is what one would hope for as an analogue of the moduli properties of \(\bar M_{0,n}\). One would also hope to be able to complete a 1-parameter family (say over a punctured disc, over the complex numbers) in a natural way. One of the main results of this paper is that this is indeed possible: the necessary description of the central fibre is similar to the one given by Kapranov for \(\bar M_{0,n}\) and involves combinatorics controlled by the Bruhat-Tits building of \(\text{ PGL}_r\)
By analogy with the case of \(\bar M_{0,n}\) one could also hope that the normalisation of \(\bar X(r,n)\) would be the log canonical model. The authors show, however, that this is not the case: on the contrary, already for \(r=3\) the singularities that can occur are essentially arbitrary. This is slightly disappointing but perhaps not wholly surprising, and leads the authors naturally to the question of what the log canonical compactification is. They conjecture that it is \(\bar X(r,n)\) in the few cases where they cannot show that it is not (there is an exception in characteristic 2).
The proofs form part of a rather detailed study of of Chow quotients in general and \(\bar X(r,n)\) in particular, which is really the content of the paper. This is not readily summarised, especially as in order to keep the length of the paper within reasonable bounds the authors have not repeated material from Kapranov’s and Lafforgue’s papers.

Reviewer: G. K. Sankaran (Bath)

MSC:

14J10	Families, moduli, classification: algebraic theory
14H10	Families, moduli of curves (algebraic)
14E05	Rational and birational maps
14L30	Group actions on varieties or schemes (quotients)
52C35	Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Keywords:

Chow quotient; compactification; log canonical model; families

Citations:

Zbl 0790.14020; Zbl 0965.14024

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