Cohomological dimension and arithmetical rank of some determinantal ideals

TL;DR

This paper computes the cohomological dimension and arithmetical rank of 2-minor ideals of certain structured matrices, extending previous results to new matrix forms and mixed cases, revealing these invariants are often smaller than those of generic matrices.

Contribution

It extends the computation of cohomological dimension and arithmetical rank to matrices with Jordan and mixed blocks, generalizing prior work on scroll blocks.

Findings

Computed cd and ara for matrices with Jordan blocks

Extended results to matrices with mixed Jordan and scroll blocks

Found ara is less than for generic matrices in studied cases

Abstract

Let $M$ be a $(2 \times n)$ 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 $(2 \times n)$-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. B\u{a}descu and Valla computed $\mathrm{ara}(I_2(M))$ when $M$ is a concatenation of scroll blocks. In this case we compute $\mathrm{cd}(I_2(M))$ and extend these results to concatenations of Jordan blocks. Eventually we compute $\mathrm{ara}(I_2(M))$ and $\mathrm{cd}(I_2(M))$ in an interesting mixed case, when $M$ contains both Jordan and scroll blocks. In all cases we show that $\mathrm{ara}(I_2(M))$ is less than the arithmetical rank of the determinantal ideal of a generic matrix.