The moduli space of cubic fourfolds
Author:
Radu Laza
Journal:
J. Algebraic Geom. 18 (2009), 511-545
DOI:
https://doi.org/10.1090/S1056-3911-08-00506-7
Published electronically:
June 5, 2008
MathSciNet review:
2496456
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We describe the GIT compactification of the moduli space of cubic fourfolds (cubic hypersurfaces in the five dimensional projective space), with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.
References
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References
- D. Allcock, The moduli space of cubic threefolds, J. Algebraic Geom. 12 (2003), no. 2, 201–223. MR 1949641 (2003k:14043)
- ---, Personal communication (2007).
- D. Allcock, J. A. Carlson, and D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom. 11 (2002), no. 4, 659–724. MR 1910264 (2003m:32011)
- ---, The Moduli Space of Cubic Threefolds as a Ball Quotient, arXiv:math/ 0608287v1 [math.AG] (2006), 77 pp.
- P. Aluffi, Singular schemes of hypersurfaces, Duke Math. J. 80 (1995), no. 2, 325–351. MR 1369396 (97b:14057)
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
- V. I. Arnol$’$d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser, Boston, MA, 1985.
- D. Avritzer and R. Miranda, Stability of pencils of quadrics in $\textbf {P}^ 4$, Bol. Soc. Mat. Mexicana (3) 5 (1999), no. 2, 281–300. MR 1738422 (2000j:14014)
- A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706. MR 818549 (87c:14047)
- W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. MR 1153249 (93a:20069)
- P. A. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994. MR 1288523 (95d:14001)
- M. Gross and S. Popescu, Equations of $(1,d)$-polarized abelian surfaces, Math. Ann. 310 (1998), no. 2, 333–377. MR 1602020 (99d:14046)
- J. Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. MR 1416564 (97e:14001)
- B. Hassett, Special cubic fourfolds, Compositio Math. 120 (2000), no. 1, 1–23. MR 1738215 (2001g:14066)
- F. Kirwan, Moduli spaces of degree $d$ hypersurfaces in $\textbf {P}_ n$, Duke Math. J. 58 (1989), no. 1, 39–78. MR 1016413 (90k:14006)
- F. C. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), no. 1, 41–85. MR 799252 (87a:14010)
- E. Looijenga, Compactifications defined by arrangements. II. Locally symmetric varieties of type IV, Duke Math. J. 119 (2003), no. 3, 527–588. MR 2003125 (2004i:14042b)
- E. Looijenga and R. Swierstra, The period map for cubic threefolds, Compositio Math. 143 (2007), no. 4, 1037–1049. MR 2339838
- D. Luna, Adhérences d’orbite et invariants, Invent. Math. 29 (1975), no. 3, 231–238. MR 0376704 (51:12879)
- S. Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003. MR 2004218 (2004g:14002)
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 (95m:14012)
- K. G. O’Grady, Irreducible symplectic 4-folds numerically irreducible symplectic 4-folds numerically equivalent to $(K3)^{[2]}$, arXiv:math/0504434v3 [math.AG] (2005), 44 pp.
- V. L. Popov and È. B. Vinberg, Invariant theory, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 137–314, 315. MR 1100485 (92d:14010)
- T. G. Room, The Geometry of the Determinantal Loci, Cambridge University Press, 1938.
- J. Shah, A complete moduli space for $K3$ surfaces of degree $2$, Ann. of Math. (2) 112 (1980), no. 3, 485–510. MR 595204 (82j:14030)
- ---, Degenerations of $K3$ surfaces of degree $4$, Trans. Amer. Math. Soc. 263 (1981), no. 2, 271–308. MR 594410 (82g:14039)
- P. Vermeire, On the regularity of powers of ideal sheaves, Compositio Math. 131 (2002), no. 2, 161–172. MR 1898433 (2003f:14016)
- C. Voisin, Théorème de Torelli pour les cubiques de $\textbf {P}^ 5$, Invent. Math. 86 (1986), no. 3, 577–601. MR 860684 (88g:14006)
- C. T. C. Wall, Geometric invariant theory of linear systems, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 1, 57–62. MR 684274 (84m:14012)
- M. Yokoyama, Stability of cubic hypersurfaces of dimension 3 and 4, Preprint Nagoya University.
- ---, Stability of cubic 3-folds, Tokyo J. Math. 25 (2002), no. 1, 85–105. MR 1908216 (2003e:14031)
Additional Information
Radu Laza
Affiliation:
Department of Mathematics, University of Michigan, 3863 East Hall, Ann Arbor, Michigan 48109
MR Author ID:
692317
ORCID:
0000-0001-9631-1361
Email:
rlaza@umich.edu
Received by editor(s):
March 7, 2007
Received by editor(s) in revised form:
October 10, 2007
Published electronically:
June 5, 2008
