Compactification of the moduli space of hyperplane arrangements
Authors:
Paul Hacking, Sean Keel and Jenia Tevelev
Journal:
J. Algebraic Geom. 15 (2006), 657-680
DOI:
https://doi.org/10.1090/S1056-3911-06-00445-0
Published electronically:
May 2, 2006
MathSciNet review:
2237265
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Abstract |
References |
Additional Information
Abstract: Consider the moduli space $M^0$ of arrangements of $n$ hyperplanes in general position in projective $(r-1)$-space. When $r=2$ the space has a compactification given by the moduli space of stable curves of genus $0$ with $n$ marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs $(S,B)$ consisting of a variety $S$ (possibly reducible) and a divisor $B=B_1+\dots +B_n$, satisfying various additional conditions. We identify the closure of $M^0$ in the moduli space of stable pairs as Kapranov’s Hilbert quotient compactification of $M^0$, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.
- Valery Alexeev, Moduli spaces $M_{g,n}(W)$ for surfaces, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 1–22. MR 1463171, DOI https://doi.org/10.1006/jcat.1996.0357
[Alexeev96b]Alexeev96b V. Alexeev, Log canonical singularities and complete moduli of stable pairs, preprint alg-geom/9608013.
- Valery Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708. MR 1923963, DOI https://doi.org/10.2307/3062130
- Robert Friedman, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), no. 1, 75–114. MR 707162, DOI https://doi.org/10.2307/2006955
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
[GS87]GS87 I. Gel’fand, V. Serganova, Combinatorial geometries and torus strata on homogeneous compact manifolds, Russian Math. Surveys 42 (1987), no. 2, 133–168.
- Paul Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), no. 2, 213–257. MR 2078368, DOI https://doi.org/10.1215/S0012-7094-04-12421-2
- Mark Haiman and Bernd Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), no. 4, 725–769. MR 2073194, DOI https://doi.org/10.1090/S1056-3911-04-00373-X
- M. M. Kapranov, Chow quotients of Grassmannians. I, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR 1237834
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Yujiro Kawamata and Yoshinori Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), no. 3, 395–409. MR 1296351, DOI https://doi.org/10.1007/BF01231538
[KT04]KT04 S. Keel and J. Tevelev, Chow quotients of Grassmannians II, preprint math.AG/ 0401159.
- J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI https://doi.org/10.1007/BF01389370
- L. Lafforgue, Chirurgie des grassmanniennes, CRM Monograph Series, vol. 19, American Mathematical Society, Providence, RI, 2003 (French). MR 1976905
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
- Richard Stanley, Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 187–213. MR 951205, DOI https://doi.org/10.2969/aspm/01110187
- Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949
- Neil L. White, The basis monomial ring of a matroid, Advances in Math. 24 (1977), no. 3, 292–297. MR 437366, DOI https://doi.org/10.1016/0001-8708%2877%2990060-3
[Alexeev96a]Alexeev96a V. Alexeev, Moduli spaces $M_{g,n}(W)$ for surfaces, Higher-dimensional complex varieties (Trento, 1994), 1–22, de Gruyter (1996).
[Alexeev96b]Alexeev96b V. Alexeev, Log canonical singularities and complete moduli of stable pairs, preprint alg-geom/9608013.
[Alexeev02]Alexeev02 V. Alexeev, Complete moduli in the presence of semi-abelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708.
[Friedman83]Friedman83 R. Friedman, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), no. 1, 75–114.
[Fulton93]Fulton93 W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, P.U.P. (1993).
[GS87]GS87 I. Gel’fand, V. Serganova, Combinatorial geometries and torus strata on homogeneous compact manifolds, Russian Math. Surveys 42 (1987), no. 2, 133–168.
[Hacking04]Hacking04 P. Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), 213–257.
[HS04]HS04 M. Haiman and B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), 725–769.
[Kapranov93]Kapranov93 M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (1993), 29–110.
[Kato89]Kato89 K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, 1988), 191–224, Johns Hopkins Univ. Press (1989).
[KN94]KN94 Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), no. 3, 395–409.
[KT04]KT04 S. Keel and J. Tevelev, Chow quotients of Grassmannians II, preprint math.AG/ 0401159.
[KSB88]KSB88 J. Kollár and N. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338.
[Lafforgue03]Lafforgue03 L. Lafforgue, Chirurgie des Grassmanniennes, CRM Monograph Series, 19, AMS (2003).
[Matsumura89]Matsumura89 H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math., 8, C.U.P. (1986).
[Oda88]Oda88 T. Oda, Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. (3), 15, Springer (1988).
[Stanley87]Stanley87 R. Stanley, Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., vol. 11, North-Holland (1987), 187–213.
[Sturmfels95]Sturmfels95 B. Sturmfels, Gröbner bases and convex polytopes, Univ. Lecture Notes, 8, AMS (1996).
[White77]White77 N. White, The basis monomial ring of a matroid, Advances in Math. 24 (1977), 292–297.
Additional Information
Paul Hacking
Affiliation:
Department of Mathematics, Yale University, P.O. Box 208283, New Haven, Connecticut 06520
MR Author ID:
737867
Email:
paul.hacking@yale.edu
Sean Keel
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
MR Author ID:
289025
Email:
keel@math.utexas.edu
Jenia Tevelev
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
MR Author ID:
607263
Email:
tevelev@math.utexas.edu
Received by editor(s):
February 9, 2005
Received by editor(s) in revised form:
June 7, 2005
Published electronically:
May 2, 2006
Additional Notes:
The second author was partially supported by NSF grant DMS-9988874
