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On the Gauss maps of singular projective varieties

Part of: Local theory Special varieties Projective and enumerative geometry

Published online by Cambridge University Press:  09 April 2009

E. Ballico
E. Ballico
Affiliation:
Department of Mathematics, University of Trento, 38050 Povo (TN), Italy e-mail: [email protected]
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Abstract

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Here we study the dimension δ(m, X) of the general fibers of the m-Gaussian map of a singular n-dimensional variety X ⊂ Pn. We show that for all integers a, b, c, d with n ≦ a < b ≦ c < d ≦ N − 1 and a + d = b + c we have δ (a, X) + δ(d, X) > δ(b, X) + δ(c, X). If δ(X, N − 1) is very large we give some classification results which extend to the singular case some results of Ein.

Keywords

Gauss mapprojective varietyGrassmannianGauss map for singular varietiestangencycontact locusdual varietyadjunction mapping

MSC classification

Secondary: 14N05: Projective techniques 14M15: Grassmannians, Schubert varieties, flag manifolds 14B05: Singularities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 119 - 130
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Beltrametti, M. and Sommese, A. J., The adjunction theory of complex projective varieties (Walter de Gruyter, Berlin, 1995).CrossRefGoogle Scholar
[2]Ein, L., ‘Varieties with small dual varieties, II’, Duke Math. J. 52 (1985), 895–907.CrossRefGoogle Scholar
[3]Ein, L., ‘Varieties with small dual varieties, I’, Invent. Math. 86 (1986), 63–74.CrossRefGoogle Scholar
[4]Fujita, T., ‘Remarks on quasi-polarized varieties’, Nagoya Math. J. 115 (1989), 105–123.CrossRefGoogle Scholar
[5]Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants (Birkhäuser, Boston, 1994).CrossRefGoogle Scholar
[6]Hefez, A. and Kleiman, S. L., ‘Notes on duality for projective varieties’, in: Geometry today (eds. Arbarello, E., Processi, C. and Strickland, E.), Progr. Math. 60 (Birkhäuser, Boston, 1985) pp. 143–185.Google Scholar
[7]Johnson, M. L., Higher secant varieties (Ph.D. Thesis, Brandeis University, 1994).Google Scholar
[8]Jouanolou, J.-P., Théorémes de Bertini et applications, Progr. Math. 42 (Birkhäuser, Boston, 1983).Google Scholar
[9]Kaji, H., ‘On the inseparable degrees of the Gauss map and the projection of the conormal variety to the dual of higher order for space curves’, Math. Ann. 292 (1992), 529–532.CrossRefGoogle Scholar
[10]Kleiman, S. L., ‘The enumerative theory of singularities’, in: Real and Complex Singularities, Oslo, 1976 (ed. Holm, P.) (Sijitoff and Noordhoof, Alphen aan den Rijn, 1977) pp. 297–396.Google Scholar
[11]Kleiman, S. L., ‘Tangency and duality’, in: Proc. 1984 Vancouver Conference in Algebraic Geometry, CMS-AMS Conference Proceedings 6 (Amer. Math. Soc., Providence, RI, 1986).Google Scholar
[12]Kleiman, S. L. and Piene, R., ‘On the inseparability of the Gauss map’, in: Enumerative Algebraic Geometry, Proc. Zeuthen Symposium, Contemp. Math. 123 (Amer. Math. Soc., Providence, RI, 1991) pp. 149–160.CrossRefGoogle Scholar
[13]Landsberg, J., ‘Algebraic geometry and projective differential geometry—Seoul National University concentrated lecture series’, preprint math.AG/9809184, 1997.Google Scholar
[14]Tango, H., ‘On (n − 1)-dimensional projective spaces contained in the Grassmann variety Gr(n, 1)’, J. Math. Kyoto Univ. 14 (1974), 415–460.Google Scholar
[15]Tango, H., ‘On morphisms from projective space Pn’, J. Math. Kyoto Univ. 16 (1976), 201–207.Google Scholar
[16]Zak, F. L., Tangents and secants of algebraic varieties, Transl. Math. Monographs 127 (Amer. Math. Soc., Providence, RI, 1983).Google Scholar