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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. $$X^*=X).$$
 14J40 $$n$$-folds ($$n>4$$) 14N05 Projective techniques in algebraic geometry 14M07 Low codimension problems in algebraic geometry