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On the superadditivity of secant defects. (English) Zbl 0727.14029
Let $$X^ n\subset {\mathbb{P}}^ N$$ be a projective algebraic variety. The k-th secant variety $$S^ kX$$ is defined as closure of the union of k- dimensional linear subspaces spanned by general collections of $$k+1$$ points of X. The invariants $$s_ k=\dim (S^ kX)$$, $$k\leq k_ 0$$, where $$k_ 0=\min \{r| S^ rX={\mathbb{P}}^ N\}$$ yield important information about the extrinsic geometry of the variety X. It is more convenient to work with another set of invariants $$\delta_ k=n+s_{k-1}-s_ k+1$$; $$\delta_ k$$ is called the k-th secant defect of X (the secant variety $$S^ rX$$ has maximal possible dimension $$s_ r=(r+1)n+r$$ if and only if $$\delta_ k=0$$ for all $$k\leq r)$$. As a useful tool in the study of linear systems of hyperplane sections on varieties of small codimension the reviewer stated the following theorem [cf. Funct. Anal. Appl. 19, 165-173 (1985); translation from Funkts. Anal. Prilozh. 19, No.3, 1-10 (1985; Zbl 0577.14032)]:
If a variety X is nonsingular and $$\delta =\delta_ 1>0$$, then the function $$\delta_ k$$ is superadditive in the interval $$[0,k_ 0].$$
Actually the reviewer used only a corollary of this theorem, viz. $$\delta_ k\geq k\delta$$ for $$k\leq k_ 0$$. Later it turned out that the proof of the above theorem relies on an unproven claim; however the above corollary and all other results of the reviewer’s paper cited above (including the bound on $$h^ 0(X,{\mathcal O}_ X(1))$$ in terms of N-n) are valid. Ådlandsvik constructed an example of a surface for which the function $$\delta_ k$$ is not superadditive, however in this example $$\delta =\delta_ 1=0$$. At present it is still an open problem whether or not the superadditivity theorem holds in full generality. However there are indications that even if we do not assume that $$\delta >0$$ Ådlansvik’s example might be the only exception to this theorem. - In the paper under review the author proves the following special case of the superadditivity theorem.
Let $$\ell +m=k\leq k_ 0$$, $$2m\leq k_ 0$$. Then $$\delta_ k\geq \delta_{\ell}+\delta_ m$$ provided that the variety $$S^{m-1}X$$ is almost smooth.
According to the author, a variety $$Y^ m$$ is called almost smooth if for all points $$y\in Y$$ we have $$T'_{Y,y}\subset S(y,Y)$$, where S(y,Y) is the closure of union of chords of Y passing through y and $$T'_{Y,y}$$ is the tangent star to Y at the point y, i.e. the union of limits of chords of Y whose endpoints converge to y. Since a nonsingular variety is almost smooth, this result implies the above corollary.
It should be pointed out that similar problems are considered from a different viewpoint in the forthcoming paper by A. Holme and J. Roberts, “Zak’s theorem on superadditivity” to appear in the Proc. of Bergen Conference on algebraic geometry).
Reviewer: F.L.Zak (Moskva)

##### MSC:
 14N05 Projective techniques in algebraic geometry 14E25 Embeddings in algebraic geometry 51N35 Questions of classical algebraic geometry
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##### References:
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