Abstract
We investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems. The first generalizes Clemens’ famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of Ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties. Both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety. We also obtain an infinitesimal Torelli theorem for hypersurfaces of toric varieties.
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References
Batyrev V.: On the classification of smooth projective toric varieties. Tohoku Math. J. 43, 569–585 (1991)
Batyrev V., Cox D.: On the Hodge structure of projective hypersurfaces in toric varieties. Duke Math. J. 75, 293–338 (1994)
Clemens H.: Curves on generic hypersurfaces. Ann. Sci. École Norm. Sup. 19, 626–636 (1986)
Cox D.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995)
Di Rocco S.: Generation of k-jets on toric varieties. Math. Z. 231, 169–188 (1999)
Ein L.: Subvarieties of generic complete intersections. II. Math. Ann. 289, 465–471 (1991)
Fakhruddin, N.: Multiplication maps of linear systems on smooth projective toric surfaces. arXiv: math. AG/0208178
Fakhruddin, N.: Zero cycles on generic hypersurfaces of large degree. arXiv: math. AG/0208179
Kleinschmidt P.: A classification of toric varieties few generators. Aequationes Math. 35, 254–266 (1988)
Mavlyutov A.: Semiample hypersurfaces in toric varieties. Duke Math. J. 101, 85–116 (2000)
Mustaţǎ M.: Vanishing theorems on toric varieties. Tohoku Math. J. 54, 451–470 (2002)
Oda, T.: Convex Bodies and Algebraic Geometry—An Introduction to the Theory of Toric Varieties. Ergebnisse der Math. (3) 15. Springer, Berlin (1988)
Oda, T.: Problems on Minkowski sums of convex lattice polytopes. preprint (1997)
Reid, M. : Decomposition of toric morphisms. Arithmetic and Geometry, vol. II, pp. 395–418. Progr. Math., vol. 36. Birkhäuser Boston (1983)
Voisin C.: Variations de structure de Hodge et zéro-cycles sur les surfaces générales. Math. Ann. 299, 77–103 (1994)
Voisin C.: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44, 200–214 (1996)
Voisin C.: A correction on a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 49, 601–611 (1998)
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Ikeda, A. Subvarieties of generic hypersurfaces in a nonsingular projective toric variety. Math. Z. 263, 923–937 (2009). https://doi.org/10.1007/s00209-008-0446-y
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DOI: https://doi.org/10.1007/s00209-008-0446-y