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Rank varieties of matrices. (English) Zbl 0736.14023

Commutative algebra, Proc. Microprogram, Berkeley/CA (USA) 1989, Publ., Math. Sci. Res. Inst. 15, 173-212 (1989).
[For the entire collection see Zbl 0721.00007.]
The authors study varieties of square matrices defined by conditions on the ranks of polynomial functions on them. Let \(V\) be an \(n\)-dimensional vector space over the complex numbers, \(X=\text{End}(V)\). A rank set is a set of the form \(\{A\in X\mid \text{rank}(p_ i(A))\leq r_ i, i\geq 1\}\) where \(p_ i(t)\) are polynomials of one variable and \(r_ i\) integer numbers. For any sequence of complex numbers \(\lambda=(\lambda_ 1,\ldots,\lambda_ s)\) and for any doubly indexed set of integers \(r(i,j)\), \(i=1,\ldots,s, j\geq1,\) let \(X_{r,\lambda}=\{A\in X\mid \text{rank}(A-\lambda_ i)^ j\leq r(i,j), 1\leq i\leq s, 1\leq j\}\). It is shown that the \(X_{r,\lambda}\) are the irreducible components of rank sets and that they are always Gorenstein with rational singularities. The authors also compute tangent spaces and limits of them under deformations of the defining polynomial functions. Finally, the singular loci of the hypersurfaces given in the space of \(n\times n\) matrices by the vanishing of a single coefficient of the characteristic polynomial is computed.

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
15A03 Vector spaces, linear dependence, rank, lineability
14M12 Determinantal varieties

Citations:

Zbl 0721.00007
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