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

Abstract

Let R=k[T1,…,Tf] be a standard graded polynomial ring over the field k and Ψ be an f×g matrix of linear forms from R, where 1≤g<f. Assume [T1⋯Tf]Ψ is 0 and that gradeIg(Ψ) is exactly one short of the maximum possible grade. We resolve R‾=R/Ig(Ψ), prove that R‾ has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R‾, and prove that if f−g is even, then the ideal Ig(Ψ) is unmixed. Furthermore, if f−g is odd, then we identify an explicit generating set for the unmixed part, Ig(Ψ)unm, of Ig(Ψ), resolve R/Ig(Ψ)unm, and record explicit formulas for the h-vector of R/Ig(Ψ)unm. (The rings R/Ig(Ψ) and R/Ig(Ψ)unm automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.