zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] ÅDLANDSVIK (B.) . - A characterisation of Veronese varieties by higher secant varieties , Preprint University of Bergen, 1984 . [2] ÅDLANDSVIK (B.) . - Letter to F.L. Zak , September 1986 . [3] FULTON (W.) , LAZARSFELD (R.) . - Connectivity and its applications to algebraic geometry , Algebraic Geometry. pp. 26-92, Berlin, Springer-Verlag, 1978 , (Lecture Notes in Math., 862). MR 83i:14002 | Zbl 0484.14005 · Zbl 0484.14005 [4] HARTSHORNE (R.) . - Varieties of small codimension in projective space , Bull. Amer. Math. Soc., t. 80, 1974 , p 1017-1032. Article | MR 52 #5688 | Zbl 0304.14005 · Zbl 0304.14005 · doi:10.1090/S0002-9904-1974-13612-8 · minidml.mathdoc.fr [5] JOHNSON (K.W.) . - Immersion and embedding of projective varieties , Acta Math., t. 140, 1978 , p 49-74. MR 57 #3120 | Zbl 0373.14005 · Zbl 0373.14005 · doi:10.1007/BF02392303 [6] TERRACINI (A.) . - Sulle Vk per cui la varietà degli Sh (h + 1)-seganti ha dimensione minore dell’ordinario , Rend. Circ. Mat. Palermo, t. 31, 1911 , p 392-396. JFM 42.0673.02 · JFM 42.0673.02 [7] WHITNEY (H.) . - Complex Analytic Varieties . - USA, Addison-Wesley, 1972 . MR 52 #8473 | Zbl 0265.32008 · Zbl 0265.32008 [8] ZAK (F.L.) . - Projections of algebraic varieties , Math. USSR Sb., t. 44, 1983 , p 535-544. Zbl 0511.14026 · Zbl 0511.14026 · doi:10.1070/SM1983v044n04ABEH000986 [9] ZAK (F.L.) . - Linear systems of hyperplane sections on varieties of low codimension , Functional Anal. Appl., t. 19, 1985 , p 165-173. MR 87d:14040 | Zbl 0577.14032 · Zbl 0577.14032 · doi:10.1007/BF01076616
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.