Cohomological dimension and arithmetical rank of some determinantal ideals

Abstract

Let M be a (2 × n) non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal I_2(M) generated by the 2-minors of M. Over an algebraically closed field, any (2×n)-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Badescu and Valla computed ara(I_2 (M)) when M is a concatenation of scroll blocks. In this case we compute cd(I2 (M)) and extend these results to concatenations of Jordan blocks. Eventually we compute ara(I_2(M)) and cd(I_2 (M)) in an interesting mixed case, when M contains both Jordan and scroll blocks. In all cases we show that ara(I_2(M)) is less than the arithmetical rank of the determinantal ideal of a generic matrix.

Keywords

• ideals of minors
• cohomological dimension
• arithmetical rank