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Linear sections of determinantal varieties. (English) Zbl 0681.14028
The author develops an elegant general approach in order to study a genericity property of determinantal varieties, which has several applications. Regarding a surjective pairing \(\mu:\quad A\otimes B\to C\) of finite dimensional vector spaces over an algebraically closed field as a subspace M of \(Hom(A^*,B)\), the author says that M is k-generic if the kernel of \(\mu\) does not contain any sums of \(\leq k\) pure tensors \(a\otimes b\). In this paper the author demonstrates the effectiveness of the k-genericity property in studying linear systems on a projective variety.
In order to formulate the first main result of the paper, we need some basic notions. Denote \(H=Hom(A^*,B)\). For any subspace \(M\subset H\) with \(\dim (M)=m\) and for any \(k=0,...,m\) we write \(M_ k\) for the locus of maps in M of rank \(\leq k\). The author says that M meets \(H_ k\) properly if \(co\dim_ M(M_ k)=co\dim_ H(H_ k)\). - Without any loss of generality we may assume \(w:=\dim (B)\leq \dim (A).\)
Resiliency theorem. If \(M'\subseteq H\) is a (w-k)-generic space and \(M\subseteq M'\) is an arbitrary subspace then: \((1)\quad If\) \(co\dim_{M'}(M)\leq k\), then M meets \(H_ k\) properly. \((2)\quad If\) \(co\dim_{M'}(M)\leq k-1\), then \(M_ k\) is reduced and irreducible. \((3)\quad If\) \(k<w-1\) and \(co\dim_{M'}(M)\leq k-2\), then \(M_ k\) is normal.
The second main result is a classification theorem which lists the most degenerate cases in the previous theorem.
Reviewer: V.V.Iliev

14M12 Determinantal varieties
14C20 Divisors, linear systems, invertible sheaves
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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