On determinantal ideals and algebraic dependence

Abstract

Let X be a matrix with entries in a polynomial ring over an algebraically closed field K. We prove that, if the entries of X outside some (t × t)-submatrix are algebraically dependent over K, the arithmetical rank of the ideal It(X) of t-minors of X drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by k if X has k zero entries. This upper bound turns out to be sharp if char K = 0, since it then coincides with the lower bound provided by the local cohomological dimension. (2010): 13A15, 14M10, 14M12, 13D45. Mathematics Subject Classification

Local cohomology -- an invitation

“…If K is algebraically closed, Barile and Macchia study in [BM19] the number of elements needed to generate the ideal of t-minors of a matrix X up to radical, if the entries of X outside some fixed t ˆt-submatrix are algebraically dependent over K. They prove that the this number drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by k if X has k zero entries. Notation 3.13.…”