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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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