zbMATH — the first resource for mathematics

Varieties with small dual varieties. I. (English) Zbl 0603.14025
Let $$X\subset {\mathbb{P}}^ N$$ be a projective complex manifold of dimension n, and let $$X^*\subset {\mathbb{P}}^{N*}$$, dim $$X^*=n^*$$ be the dual variety formed by the points corresponding to the tangent hyperplanes. As a rule, $$n^*=N-1$$, i.e. def X$$=N-n^*-1=0$$. The paper under review is devoted to varieties X for which def X$$>0$$. The reviewer has shown that for all $$(smooth)\quad X: n^*\geq n.$$ The main purpose of the present paper is to classify varieties for which $$n^*=n$$ under the assumption that $$n\leq 2N/3$$ (we recall that from Hartshorne’s conjecture it follows that, for $$n>2N/3,\quad def X=0).$$
The main auxiliary result having also many other applications consists in a description of the structure of the normal bundle $$N_{L/X}$$, where L is a linear subspace of dimension def X along which a generic hyperplane from $$X^*$$ is tangent to X (this description also yields some old results, e.g. the reviewer’s theorem to the effect that def $$X\leq n-2$$ and Landman’s theorem according to which def $$X\equiv n (mod 2)$$ if $$n^*<N-1)$$. The author’s study based, besides the above results, on the Bejlinson spectral sequence shows that the only varieties for which $$n\leq 2N/3$$, $$n^*=n$$ are the hypersurfaces, the Segre varieties $${\mathbb{P}}^ 1\times {\mathbb{P}}^{n-1}\subset {\mathbb{P}}^{2n-1}$$, $$n\geq 3$$, the Grassmann variety $$G(4,1)^ 6\subset {\mathbb{P}}^ 9$$, and the spinor variety $$S^{10}\subset {\mathbb{P}}^{15}$$ (all these varieties, with the exception of hypersurfaces of degree greater $$than^ 2,$$ are self-dual, i.e. $$X^*=X).$$
Reviewer: F.L.Zak

MSC:
 14J40 $$n$$-folds ($$n>4$$) 14N05 Projective techniques in algebraic geometry 14M07 Low codimension problems in algebraic geometry
Full Text:
References:
 [1] Ballico, E., Chiantini, L.: On smooth subcanonical varieties of codimension 2? n?4 (To appear in Ann. Mat. Pure Appl.) · Zbl 0549.14015 [2] Barth, W.: Transplanting cohomology class in complex projective space. Am. J. Math.92, 951-961 (1970) · Zbl 0206.50001 · doi:10.2307/2373404 [3] Ein, L.: Stable vector bundle on projective space in charp>0. Math. Ann.254, 53-72 (1980) · Zbl 0437.14007 · doi:10.1007/BF01457885 [4] Ein, L.: Varieties with small dual varieties II. (preprint) · Zbl 0603.14026 [5] Elencwajg, G., Hirschowitz, A., Schneider, M.: Les fibres uniformes de rang au plusn sur? n (?). Proceedings of the Nice Conference 1979 on Vector bundles and Differential equations · Zbl 0456.32009 [6] Fulton, W., Lazarsfeld, R.: Connectivity and its applications in algebraic geometry, Lect. Notes Math.862, 26-92 (1981) · Zbl 0484.14005 · doi:10.1007/BFb0090889 [7] Fujita, T.: On the structure of polarized manifolds with total deficiency one I. J. Math. Soc. Jpn32-4, 709-775 (1980) · Zbl 0474.14017 · doi:10.2969/jmsj/03240709 [8] Fujita, T.: On the structure of polarized manifolds with total deficiency one II. J. Math. Soc. Jpn.33-3, 415-434 (1981) · Zbl 0474.14018 · doi:10.2969/jmsj/03330415 [9] Fujita, T., Roberts, J.: Varieties with small secant varieties: the extremal case. Am. J. Math.103, 953-976 (1981) · Zbl 0475.14046 · doi:10.2307/2374254 [10] Griffiths, P., Harris, J.: Algebraic geometry and local differential geometry. Ann. Sci. Ec. Norm. Super.12, 355-432 (1979) · Zbl 0426.14019 [11] Hartshorne, R.: Varieties of low codimension in projective space. Bull. Am. Math. Soc.80, 1017-1032 (1974) · Zbl 0304.14005 · doi:10.1090/S0002-9904-1974-13612-8 [12] Hartshorne, R.: Algebraic geometry. Graduate Text in Mathematics, vol. 52. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001 [13] Hefez, A., Kleiman, S.: Notes on duality for projective varieties (to appear) · Zbl 0579.14047 [14] Ionescu, P.: An enumeration of all smooth projective varieties of degree 5 and 6. Increst Preprint Series Math.74 (1981) [15] Kleiman, S.: About the conormal scheme (to appear) · Zbl 0547.14031 [16] Kleiman, S.: Plane forms and multiple point formulas (to appear) · Zbl 0492.14044 [17] Kleiman, S.: The enumerative theory of singularities. In: Holme, P. (ed.): Real and complex singularities. Oslo 1976, pp. 297-396. Sijtoff and Noordhoof 1977 [18] Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ.13-1, 31-47 (1973) · Zbl 0261.32013 [19] Lazarsfeld, R., Van de Ven, A.: Recent work of F.L. Zak (appeared in DMV-seminar) [20] Lamothe, K.: The topology of complex projective varieties after S. Lefschetz. Topology20, 15-51 (1980) · Zbl 0445.14010 · doi:10.1016/0040-9383(81)90013-6 [21] Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math.110, 593-606 (1979) · Zbl 0423.14006 · doi:10.2307/1971241 [22] Mumford, D.: Some footnote of the work of C.P. Ramanujam. In: Ramanujam, C.P.: A Tribute, pp. 247-262. Berlin-Heidelberg-New York: Springer 1978 [23] Okonek, C., Spindler, H., Schneider, M.: Vector bundles on complex projective space. Prog. Math.3, Basel, Boston: BirkhĂ¤user (1980) · Zbl 0438.32016 [24] Ramanajam, C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Soc.36, 41-51 (1972) · Zbl 0276.32018 [25] Room, T.: A Synthesis of Clifford matrices and its generalization. Am. J. Math.74, 967-984 (1952) · Zbl 0046.24201 · doi:10.2307/2372238 [26] Sommese, A.J.: Hyperplane section of projection surface I-the adjunction mapping. Duke Math. J.46, 377-401 (1979) · Zbl 0415.14019 · doi:10.1215/S0012-7094-79-04616-7 [27] Van de Ven, A.: On the 2-connectedness of the very ample divisor on a surface. Duke Math. J.46, 403-407 (1979) · Zbl 0458.14003 · doi:10.1215/S0012-7094-79-04617-9 [28] Zak, F.: Projection of algebraic varieties. Math. U.S.S.R. Sbornik,44, 535-544 (1983) · Zbl 0511.14026 · doi:10.1070/SM1983v044n04ABEH000986 [29] Zak, F.: Varieties of small codimension arising from group action. Addendum of ?Recent work of F.L. Zak? (to appear)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.