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8 - Special prime Fano fourfolds of degree 10 and index 2

Published online by Cambridge University Press:  05 January 2015

By
O. Debarre ,
A. Iliev  and
L. Manivel
Edited by
Christopher D. Hacon ,
Mircea Mustaţă  and
Mihnea Popa
O. Debarre
Affiliation:
École Normale Supérieure et CNRS
A. Iliev
Affiliation:
Seoul National University
L. Manivel
Affiliation:
Université de Grenoble I et CNRS
Christopher D. Hacon
Affiliation:
University of Utah
Mircea Mustaţă
Affiliation:
University of Michigan, Ann Arbor
Mihnea Popa
Affiliation:
University of Illinois, Chicago
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Summary

Abstract

We analyze (complex) prime Fano fourfolds of degree 10 and index 2. Mukai gave in [M1] a complete geometric description; in particular, most of them are contained in a Grassmannian G(2, 5). As in the case of cubic fourfolds, they are unirational and some are rational, as already remarked by Roth in 1949.

We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2, 5). Special fourfolds correspond to a countable union of hypersurfaces (the Noether-Lefschetz locus) in the period domain, labelled by a positive integer d. We describe special fourfolds for some low values of d. We also characterize those integers d for which special fourfolds do exist.

Dedicated to Robert Lazarsfeld on the occasion of his sixtieth birthday

1 Introduction

One of the most vexing classical questions in complex algebraic geometry is whether there exist irrational smooth cubic hypersurfaces in P5. They are all unirational, and rational examples are easy to construct (such as Pfaffian cubic fourfolds) but no smooth cubic fourfold has yet been proven to be irrational. The general feeling seems to be that the question should have an affirmative answer but, despite numerous attempts, it is still open.

In a couple of very interesting articles on cubic fourfolds ([H1], [H2]), Hassett adopted a Hodge-theoretic approach and, using the period map (proven to be injective by Voisin in [V]) and the geometry of the period domain, a 20-dimensional bounded symmetric domain of type IV, he related geometric properties of a cubic fourfold to arithmetical properties of its period point.

We do not solve the rationality question in this paper, but investigate instead similar questions for a nother family of Fano fourfolds (see Section 2 for their definition).

Type
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Information
Recent Advances in Algebraic Geometry
A Volume in Honor of Rob Lazarsfeld’s 60th Birthday
 , pp. 123 - 155
Publisher: Cambridge University Press
Print publication year: 2015

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References

[AM] Andreatta, M. and Mella, M.Morphisms of projective varieties from the viewpoint of minimal model program. Dissertationes Math. (Rozprawy Mat.) 413 (2003), 72 pp.Google Scholar
[BH] Borel, A. and Hirzebruch, F.Characteristic classes and homogeneous spaces, I. Amer. J. Math. 80 (1958) 458–538.Google Scholar
[B] Bruinier, J.H.Hilbert modular forms and their applications. In The 1-2-3 of Modular Forms. Berlin: Springer-Verlag, pp. 105–179, 2008.
[CM] Conte, A. and Murre, J. P.The Hodge conjecture for fourfolds admitting a covering by rational curves. Math. Ann. 238 (1978) 79–88.Google Scholar
[CS] Conway, J. H. and Sloane, N. J. A.Sphere Packings, Lattices and Groups, 2nd edn. With additional contributions by E., Bannai, R.E., Borcherds, J., Leech, S.P., Norton, A.M., Odlyzko, R.A., Parker, L., Queen, and B.B., Venkov. Grundlehren der Mathematischen Wissenschaften 290, New York: Springer-Verlag, 1993.
[DIM1] Debarre, O., Iliev, A., and Manivel, L.On the period map for prime Fano threefolds of degree 10. J. Algebraic Geom. 21 (2012) 21–59.Google Scholar
[DIM2] Debarre, O., Iliev, A., and Manivel, L.On nodal prime Fano threefolds of degree 10. Sci. China Math. 54 (2011) 1591–1609.Google Scholar
Enriques, F.Sulla irrazionalita da cui puo farsi dipendere la resoluzione d'un'equazione algebrica f(x,y,z) = 0 con funzione razionali di due parametri. Math. Ann. 49 (1897) 1–23.Google Scholar
Fu, B. and Wang, C.-L.Motivic and quantum invariance under stratified Mukai flops. J. Differential Geom. 80 (2008) 261–280.Google Scholar
Gritsenko, V., Hulek, K., and Sankaran, G. K.Moduli spaces of irreducible symplectic manifolds. Compos. Math. 146 (2010) 404–434.Google Scholar
[G] Gushel', N. P.Fano varieties of genus 6 (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) 1159–1174, 1343. English transl.: Math. USSR Izv. 21 (1983) 445–459.Google Scholar
[H1] Hassett, B.Special cubic fourfolds. Compos. Math. 120 (2000) 1–23.Google Scholar
[H2] Hassett, B.Some rational cubic fourfolds. J. Algebraic Geom. 8 (1999) 103–114.Google Scholar
[HT1] Hassett, B. and Tschinkel, Y.Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces. J. Inst. Math. Jussieu 9 (2010) 125–153.Google Scholar
[HT2] Hassett, B. and Tschinkel, Y.Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11 (2001) 1201–1228.Google Scholar
[I] Iliev, A.The Fano surface of the Gushel threefold. Compos. Math. 94 (1994) 81–107.Google Scholar
[IM] Iliev, A. and Manivel, L.Fano manifolds of degree 10 and EPW sextics. Ann. Sci. École Norm. Sup. 44 (2011) 393–426.Google Scholar
[IP] Iskovskikh, V. A. and Prokhorov, Y.Fano varieties. In Algebraic Geometry, V. Encyclopaedia of Mathematical Sciences, Vol. 47. Berlin: Springer-Verlag, pp. 1–247, 1999.
[JK] Johnsen, T. and Knutsen, A. L.K3 Projective Models in Scrolls. Lecture Notes in Mathematics, Vol. 1842. Berlin: Springer-Verlag, 2004.
[L] Laszlo, Y.Théorème de Torelli générique pour les intersections complètes de trois quadriques de dimension paire. Invent. Math. 98 (1989) 247–264.Google Scholar
[La] Laza, R.The moduli space of cubic fourfolds via the period map. Ann. Math. 172 (2010) 673–711.Google Scholar
[M1] Mukai, S. Curves, K3 surfaces and Fano 3-folds of genus ≤ 10. In Algebraic Geometry and Commutative Algebra, Vol. I, pp. 357–377, Kinokuniya, Tokyo, 1988.
[M2] Mukai, S.Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989) 3000–3002.Google Scholar
[M3] Mukai, S.New development of theory of Fano 3-folds: Vector bundle method and moduli problem. Sugaku 47 (1995) 125–144.Google Scholar
[N] Nikulin, V.Integral symmetric bilinear forms and some of their applications. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111–177. English transl.: Math. USSR Izv. 14 (1980) 103–167.Google Scholar
[O] O'Grady, K.Double covers of EPW-sextics. Michigan Math. J. 62 (2013) 143–184.Google Scholar
[PV] Piontkowski, J. and Van de Ven, A.The automorphism group of linear sections of the Grassmannian G(1, N). Doc. Math. 4 (1999) 623–664.Google Scholar
[P] Prokhorov, Y.Rationality constructions of some Fano fourfolds of index 2. Moscow Univ. Math. Bull. 48 (1993) 32–35.Google Scholar
[R] Roth, L.Algebraic varieties with canonical curve sections. Ann. Mat. Pura Appl.(4) 29 (1949)91–97.Google Scholar
[S] Serre, J.-P.Cours d'Arithmétique. Le Mathématicien 2. Paris: Presses Universitaires de France, 1977.
[SB] Shepherd-Barron, N.The rationality of quintic del Pezzo surfaces–a short proof. Bull. London Math. Soc. 24 (1992) 249–250.Google Scholar
[V] Voisin, C.Théorème de Torelli pour les cubiques de P5. Invent. Math. 86 (1986) 577–601 and Erratum, Invent. Math. 172 (2008) 455–458.Google Scholar
[W] Wu, W.-T.Classes caractéristiques et i-carrés d'une variété. C. R. Acad. Sci. Paris 230 (1950) 508–511.Google Scholar
[Z] Zarhin, Y.Hodge groups of K3 surfaces. J. reine und angew. Math. 341 (1983) 193–220.Google Scholar

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