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Spaces of matrices of constant rank and uniform vector bundles. (English) Zbl 1347.15024
Summary: We consider the problem of determining \(l(r, a)\), the maximal dimension of a subspace of \(a \times a\) matrices of rank \(r\). We first review, in the language of vector bundles, the known results. Then using known facts on uniform bundles we prove some new results and make a conjecture. Finally we determine \(l(r; a)\) for every \(r\), \(1 \leq r \leq a\), when \(a \leq 10\), showing that our conjecture holds true in this range.
MSC:
15A30 Algebraic systems of matrices
15A03 Vector spaces, linear dependence, rank, lineability
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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