On the superadditivity of secant defects.

*(English)*Zbl 0727.14029Let \(X^ n\subset {\mathbb{P}}^ N\) 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).

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 |

##### Keywords:

almost smooth variety; secant variety; secant defect; small codimension; superadditivity theorem##### References:

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