A matrix of linear forms which is annihilated by a vector of indeterminates

TL;DR

This paper studies the algebraic properties of ideals generated by minors of a matrix of linear forms, providing explicit resolutions, formulas, and conditions for unmixedness, with applications to blow-up algebras.

Contribution

It explicitly resolves the quotient by the ideal, proves linearity of the resolution, and characterizes unmixedness based on the parity of f-g, advancing understanding of these algebraic structures.

Findings

R/I has a g-linear resolution.

Explicit formulas for the h-vector and multiplicity of R/I.

Identification of conditions for I to be unmixed and explicit generators for I^{unm}.

Abstract

Let R be a standard graded polynomial ring in f variables over a field and Psi be an f by g matrix of linear forms from R, where g is positive and less than f. Assume that the row vector of variables annihilates Psi and that the ideal I generated by the g by g minors of Psi has grade exactly one short of the maximum possible grade. We resolve R/I, prove that I has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R/I, and prove that if f-g is even, then the ideal I is unmixed. Furthermore, if f-g is odd, then we identify an explicit generating set for the unmixed part, I^{unm}, of I, resolve R/I^{unm}, and record explicit formulas for the h-vector of R/I^{unm}. These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.