# zbMATH — the first resource for mathematics

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

##### MSC:
 14M12 Determinantal varieties 14C20 Divisors, linear systems, invertible sheaves 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
##### Keywords:
determinantal varieties; linear systems
Full Text: