×

On second fundamental forms of projective varieties. (English) Zbl 0840.14025

Summary: The projective second fundamental form at a generic smooth point \(x\) of a subvariety \(X^n\) of projective space \(\mathbb{C} \mathbb{P}^{n+a}\) may be considered as a linear system of quadratic forms \(|II|_x\) on the tangent space \(T_x X\). We prove this system is subject to certain restrictions, including a bound on the dimension of the singular locus of any quadric in the system \(|II|_x\). (The only previously known restriction was that if \(X\) is smooth, the singular locus of the entire system must be empty.) One consequence is that smooth subvarieties with \(2(a-1)<n\) are such that their third and all higher fundamental forms are zero. This says that the infinitesimal invariants of such varieties are of the same nature as the invariants of hypersurfaces, giving further evidence towards the principle [e.g. R. Hartshorne, Bull. Am. Math. Soc. 80, 1017-1032 (1974; Zbl 0304.14005)]that smooth subvarieties of small codimension should behave like hypersurfaces.
Further restrictions on the second fundamental form occur when one has more information about the variety. In this paper we discuss additional restrictions when the variety contains a linear space and when the variety is a complete intersection. These rank restrictions should prove useful both in enhancing our understanding of smooth subvarieties of small codimension, and in bounding from below the dimensions of singularities of varieties for which local information is more readily available than global information.

MSC:

14J26 Rational and ruled surfaces
14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 0304.14005
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [BCG] Bryant, R., Chern, S., Gardner, R., Goldschmidt, Griffiths, P.: Exterior Differential Systems, Berlin Heidelberg New York: Springer 1991
[2] [C] Cartan, H.: Sur la deformation projective des surfaces. Oeuvres completes, vol. 3, Part 1, pp. 441-539. 1955
[3] [FL] Fulton, W., Lazarsfeld, R.: Connectivity and its applications in algebraic geometry. In: Libgober, A., Wagreich, P., (eds.) Algebraic Geometry (Proceedings). (Lect. Notes Math., vol. 862, pp. 26-92) Berlin Heidelberg New York: Springer 1987
[4] [GH] Griffiths, P.A., Harris, J.: Algebraic Geometry and Local Differential Geometry. Ann. Sci. Ec. Norm. Super.12, 355-432 (1979) · Zbl 0426.14019
[5] [H] Hartshorne, R.: Varieties of small codimension in projective space. Bull.80, 1017-1032 (1974) · Zbl 0304.14005
[6] [K] Kline, M.: Mathematical Thought From Ancient to Modern Times, Oxford: Oxford University Press 1972 · Zbl 0277.01001
[7] [L] Landsberg, J.M.: On Degenerate Secant and Tangential Varieties and Local Differential Geometry. (Preprint available) · Zbl 0879.14025
[8] [LV] Lazarsfeld, R., Van de Ven, A.: Topics in the Geometry of Projective Space, Recent Work of F.L. Zak. DMV Seminar. Boston Basel Stuttgart: Birkhäuser 1984
[9] [Z] Zak, F.L.: Structure of Gauss Maps. Funct. Anal. Appl.21, 32-41 (1987) · Zbl 0623.14026 · doi:10.1007/BF01077983
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.