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Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  11 August 2009

Brendan Hassett  and
Yuri Tschinkel
Brendan Hassett
Affiliation:
Department of Mathematics, Rice University, 6100 South Main Street, Houston, TX 77251-1892, USA ([email protected])
Yuri Tschinkel
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA ([email protected])
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Abstract

We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

Keywords

holomorphic symplectic varietiescones of ample and moving divisorsbirational automorphisms

MSC classification

Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 125 - 153
Copyright
Copyright © Cambridge University Press 2010

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References

1.Altman, A. B. and Kleiman, S. L., Foundations of the theory of Fano schemes, Compositio Math. 34(1) (1977), 3–47.Google Scholar
2.Amerik, E. and Voisin, C., Potential density of rational points on the variety of lines of a cubic fourfold, Duke Math. J. 145(2) (2008), 379–408.CrossRefGoogle Scholar
3.Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, Volume I, Grundlehren der Mathematischen Wissenschaften, Volume 267 (Springer, 1985).CrossRefGoogle Scholar
4.Beauville, A., Variétés Kählériennes compactes avec c 1 = 0, in Geometry of K3 surfaces: moduli and periods (Palaiseau, 1981/1982), Astérisque 126 (1985), 181–192.Google Scholar
5.Beauville, A., Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64.CrossRefGoogle Scholar
6.Beauville, A. and Donagi, R., La variété des droites d'une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I 301(301) (1985), 703–706.Google Scholar
7.Boucksom, S., Le cône Kählérien d'une variété hyperkählérienne, C. R. Acad. Sci. Paris Sér. I 333(10) (2001), 935–938.CrossRefGoogle Scholar
8.Buckley, A. and Košir, T., Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115–140.CrossRefGoogle Scholar
9.Burns, D., Hu, Y. and Luo, T., Hyperkähler manifolds and birational transformations in dimension 4, in Vector bundles and representation theory conference (Columbia, MO, 2002), Contemporary Mathematics, Volume 322, pp. 141–149 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
10.Clebsch, A., Die Geometrie auf den Flächen dritter Ordnung, J. Reine Angew. Math. 65 (1866), 359–380.Google Scholar
11.Clemens, C. H. and Griffiths, P. A., The intermediate Jacobian of the cubic threefold, Annals Math. 95 (1972), 281–356.CrossRefGoogle Scholar
12.Coray, D. F., Lewis, D. J., Shepherd-Barron, N. I. and Swinnerton-Dyer, P., Cubic threefolds with six double points, in Number Theory in Progress, Zakopane-Kościelisko, 1997, Volume 1, pp. 63–74 (de Gruyter, Berlin, 1999).Google Scholar
13.Cremona, L., Ueber die Polar–Hexaeder bei den Flächen dritter ordnung, Math. Annalen 13(13) (1878), 301–304 (reprinted in Opere, Volume 3, pp. 430–433).CrossRefGoogle Scholar
14.Debarre, O., Un contre-exemple au théorème de Torelli pour les variétés symplectiques irréductibles, C. R. Acad. Sci. Paris Sér. I 299(299) (1984), 681–684.Google Scholar
15.Dickson, L. E., Algebraic theory of the expressibility of cubic forms as determinants, with application to diophantine analysis, Am. J. Math. 43(2) (1921), 102–125.CrossRefGoogle Scholar
16.Dolgachev, I. V., Luigi Cremona and cubic surfaces, in Luigi Cremona (1830–1903), Incontro di Studio, Volume 36, pp. 55–70 (Istituto Lombardo di Scienze e Lettere, Milan, 2005; in Italian).Google Scholar
17.Fryers, M. J., The movable fan of the Horrocks–Mumford quintic, preprint (2001; available at http://arXiv.org/abs/math/0102055).Google Scholar
18.Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold, in Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, Volume 10, pp. 105–165 (North-Holland, Amsterdam, 1987).CrossRefGoogle Scholar
19.Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 2 (Results in Mathematics and Related Areas (3)) (Springer, 1984).CrossRefGoogle Scholar
20.Grassmann, H., Die stereometrischen Gleichungen dritten Grades, und die dadurch erzeugten Oberflächen, J. Reine Angew. Math. 49 (1855), 47–65.Google Scholar
21.Hassett, B., Special cubic fourfolds, Compositio Math. 120(120) (2000), 1–23.CrossRefGoogle Scholar
22.Hassett, B. and Tschinkel, Y., Rational curves on holomorphic symplectic fourfolds, Geom. Funct. Analysis 11(6) (2001), 1201–1228.Google Scholar
23.Hassett, B. and Tschinkel, Y., Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Analysis, in press.Google Scholar
24.Hunt, B., The geometry of some special arithmetic quotients, Lecture Notes in Mathematicsm, Volume 1637 (Springer, 1996).CrossRefGoogle Scholar
25.Huybrechts, D., Birational symplectic manifolds and their deformations, J. Diff. Geom. 45(3) (1997), 488–513.Google Scholar
26.Huybrechts, D., Compact hyper-Kähler manifolds: basic results, Invent. Math. 135(135) (1999), 63–113.CrossRefGoogle Scholar
27.Kapranov, M. M., Chow quotients of Grassmannians, I, in I. M. Gelfand Seminar, Advanced Soviet Mathematics, Volume 16, pp. 29–110 (American Mathematical Society,Providence, RI, 1993).CrossRefGoogle Scholar
28.Kapranov, M. M., Veronese curves and Grothendieck–Knudsen moduli space , J. Alg. Geom. 2(2) (1993), 239–262.Google Scholar
29.Kawamata, Y., On the cone of divisors of Calabi–Yau fiber spaces, Int. J. Math. 8(5) (1997), 665–687.CrossRefGoogle Scholar
30.Leung, N. C., Lagrangian submanifolds in hyper Kähler manifolds, Legendre transformation, J. Diff. Geom. 61(1) (2002), 107–145.Google Scholar
31.Looijenga, E. and Peters, C., Torelli theorems for Kähler K3 surfaces, Compositio Math. 42(42) (1980/1981), 145–186.Google Scholar
32.Markman, E., Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface, preprint arXiv:math/0601304v3.Google Scholar
33.Morrison, D. R., Compactifications of moduli spaces inspired by mirror symmetry, in Journées de géométrie algébrique d'Orsay (Orsay, 1992), Astérisque 218 (1993), 243–271.Google Scholar
34.Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77(1) (1984), 101–116.CrossRefGoogle Scholar
35.Namikawa, Y., Counter-example to global Torelli problem for irreducible symplectic manifolds, Math. Annalen 324(4) (2002), 841–845.CrossRefGoogle Scholar
36.Reid, M., Minimal models of canonical 3-folds, in Algebraic Varieties and Analytic Varieties, Tokyo, 1981, Advanced Studies in Pure Mathematics, Volume 1, pp. 131–180 (North-Holland, Amsterdam, 1983).CrossRefGoogle Scholar
37.Schröter, H. E., Nachweis der 27 Geraden auf der allgemeinen Oberflächen dritter Ordnung, J. Reine Angew. Math. 62 (1863), 265–280.Google Scholar
38.Segre, B., On the rational solutions of homogeneous cubic equations in four variables, Math. Notae 11 (1951), 1–68.Google Scholar
39.Segre, C., Sulle varietà cubiche dello spazio a quattro dimensioni e su certi sistemi di rette e certe superficie dello spazio ordinario, in Memorie Reale Accademia Scienze Torino 39 (1887), 3–48 (reprinted in Opere, Volume 4 (Unione Matematica Italiana, Edizioni Cremonese, Rome, 1963).Google Scholar
40.Wierzba, J. and Wiśniewski, J. A., Small contractions of symplectic 4-folds, Duke Math. J. 120(1) (2003), 65–95.CrossRefGoogle Scholar