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

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.

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) |

PDF
BibTeX
XML
Cite

\textit{S. Keel} and \textit{J. Tevelev}, Duke Math. J. 134, No. 2, 259--311 (2006; Zbl 1107.14026)

### References:

[1] | V. Alexeev, Log canonical singularities and complete moduli of stable pairs , |

[2] | D. Barlet, “Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension Finie” in Fonctions de plusieurs variables complexes, II , Lecture Notes in Math. 482 , Springer, Berlin, 1975, 1–158. · Zbl 0331.32008 |

[3] | K. S. Brown, Buildings , reprint of 1989 original, Springer Monogr. Math., Springer, New York, 1998. |

[4] | M. Cailotto, Algebraic connections on logarithmic schemes , C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 1089–1094. · Zbl 1074.14513 |

[5] | I. V. Dolgachev, “Abstract configurations in algebraic geometry” in The Fano Conference (Torino, 2002) , Univ. Torino, Turin, 2004, 423–462. · Zbl 1068.14059 |

[6] | I. V. Dolgachev and Y. Hu, Variation of geometric invariant theory quotients , Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. · Zbl 1001.14018 |

[7] | V. G. Drinfeld, Elliptic modules (in Russian), Mat. Sb. (N.S.) 94 ( 136 ) (1974), 594–627., 656; English translation in Math. USSR-Sb. 23 (1974), 561–592. |

[8] | G. Faltings, “Toroidal resolutions for some matrix singularities” in Moduli of Abelian Varieties (Texel Island, Netherlands, 1999) , Prog. Math. 195 , Birkhäuser, Basel, 2001, 157–184. · Zbl 1028.14002 |

[9] | R. Friedman, Global smoothings of varieties with normal crossings , Ann. of Math. (2) 118 (1983), 75–114. JSTOR: · Zbl 0569.14002 |

[10] | W. Fulton, Introduction to Toric Varieties , Ann. of Math. Stud. 131 , Princeton Univ. Press, Princeton, 1993. · Zbl 0813.14039 |

[11] | I. M. Gelfand, R. M. Goresky, R. D. Macpherson, and V. V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells , Adv. in Math. 63 (1987), 301–316. · Zbl 0622.57014 |

[12] | I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants , Math. Theory Appl., Birkhäuser, Boston, 1994. · Zbl 0827.14036 |

[13] | P. Hacking, Compact moduli of hyperplane arrangements , · Zbl 1060.14034 |

[14] | P. Hacking, S. Keel, and J. Tevelev Compactification of the moduli space of hyperplane arrangements , to appear in J. Algebraic Geom. · Zbl 1117.14036 |

[15] | D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination , Chelsea, New York, 1952. · Zbl 0047.38806 |

[16] | M. M. Kapranov, “Chow quotients of Grassmannians, I” in I. M. Gelfand Seminar , Adv. Soviet Math. 16 , Part 2, Amer. Math. Soc., Providence, 1993, 29–110. · Zbl 0811.14043 |

[17] | -, Veronese curves and Grothendieck-Knudsen moduli space \(\oM_0,n\) , J. Algebraic Geom. 2 (1993), 239–262. · Zbl 0790.14020 |

[18] | M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties , Math. Ann. 290 (1991), 643–655. · Zbl 0762.14023 |

[19] | Y. Kawamata, K. Matsuda, and K. Matsuki, “Introduction to the minimal model program” in Algebraic Geometry (Sendai, Japan, 1985) , Adv. Stud. Pure Math. 10 , North-Holland, Amsterdam, 1987, 283–360. · Zbl 0672.14006 |

[20] | Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothings of degenerate Calabi-Yau varieties , Invent. Math. 118 (1994), 395–409. · Zbl 0848.14004 |

[21] | S. Keel and J. Mckernan, Contractible extremal rays of \(\overlineM_0,n\) , · Zbl 1322.14050 |

[22] | J. KolláR (with 14 coauthors), Flips and Abundance for Algebraic Threefolds (Salt Lake City, 1991) , Astérique 211 , Soc. Math. France, Montrouge, 1992. |

[23] | L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et compactification des \(\mathrm PGL_r^\mathit n+1/\mathrm PGL_r\), Invent. Math. 136 (1999), 233–271. · Zbl 0965.14024 |

[24] | -, Chirurgie des grassmanniennes , CRM Monogr. Ser. 19 , Amer. Math. Soc., Providence, 2003. |

[25] | G. A. Mustafin, Non-Archimedean uniformization , Math. USSR-Sb. 34 (1978), 187–214. · Zbl 0411.14006 |

[26] | T. Oda, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties , Ergeb. Math. Grenzgeb. (3) 15 , Springer, Berlin, 1988. · Zbl 0628.52002 |

[27] | M. C. Olsson, Logarithmic geometry and algebraic stacks , Ann. Sci. École Norm. Sup. (4) 36 (2003), 747, –791. · Zbl 1069.14022 |

[28] | -, The logarithmic cotangent complex , Math. Ann. 333 (2005), 859–931. · Zbl 1095.14016 |

[29] | E. H. Spanier, Algebraic Topology , corrected reprint of 1966 original, Springer, New York, 1981. |

[30] | D. Speyer and B. Sturmfels, The tropical Grassmannian , Adv. Geom. 4 (2004), 389–411. · Zbl 1065.14071 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.