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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. \(X^*=X).\)
[See also part II of this paper, reviewed below.]
Reviewer: F.L.Zak

MSC:
14J40 \(n\)-folds (\(n>4\))
14N05 Projective techniques in algebraic geometry
14M07 Low codimension problems in algebraic geometry
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