An alternating matrix and a vector, with application to Aluffi algebras

Abstract

Let X be a generic alternating matrix, t be a generic row vector, and J be the ideal Pf4(X)+I1(tX). We prove that J is a perfect Gorenstein ideal of grade equal to the grade of Pf4(X) plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that J defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module N which is equal to the column space of X, calculated mod Pf4(X). The module N is a self-dual maximal Cohen–Macaulay module of rank two; furthermore, J is a Bourbaki ideal for N. The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module N.