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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.
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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