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Vector spaces of matrices of low rank and vector bundles on projective spaces: An addendum to a paper by Eisenbud and Harris. (English) Zbl 0828.14009
Summary: Here we give a quick affirmative answer to a linear algebra question raised by D. Eisenbud and J. Harris [Adv. Math. 70, No. 2, 135-155 (1988; Zbl 0657.15013)]. In the quoted paper methods coming from algebraic geometry (essentially the theory of vector bundles and reflexive sheaves on $$\mathbb{P}^N)$$ were shown to be powerful tools for the study of vector spaces of matrices. Fix a positive integer $$k$$. Let $$V$$ and $$W$$ be vector spaces; set $$v : = \dim (V)$$, $$w : = \dim (W)$$, $$s : = \min (v,w)$$. Let $$M \subseteq \operatorname{Hom} (V,W)$$ be a vector space of linear maps such that the general $$f \in M$$ has rank $$k$$. Here we show essentially that if $$k$$ is fixed and “$$M$$ does not come from a simpler situation”, then $$s$$ cannot be arbitrarily large. With the notations of the quoted paper we show the existence of $$s_0 (k)$$ such that if $$s \geq s_0 (k)$$, then $$M$$ is not strongly indecomposable.
##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 15A03 Vector spaces, linear dependence, rank, lineability 14N05 Projective techniques in algebraic geometry 15A99 Basic linear algebra
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