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

TL;DR

This paper studies a special ideal generated by a Pfaffian and a linear form, proving its Gorenstein property, and explores its algebraic properties and connections to algebraic geometry, especially Grassmannians.

Contribution

It establishes the Gorenstein property of a specific ideal related to Pfaffians and linear forms, and links it to modules and algebraic geometry applications.

Findings

The ideal J is perfect Gorenstein of a specific grade.

J defines a domain, normal ring, or UFD if the base ring has these properties.

The module N is a self-dual maximal Cohen-Macaulay module of rank two.

Abstract

Let $\mathbf X$ be a generic alternating matrix, $\mathbf t$ be a generic row vector, and $J$ be the ideal $\operatorname{Pf}_4({\mathbf X})+I_1({\mathbf {t X}})$. We prove that $J$ is a perfect Gorenstein ideal of grade equal to the grade of $\operatorname{Pf}_4({\mathbf X})$ 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 $\mathcal N$ which is equal to the column space of $\mathbf X$, calculated mod $\operatorname{Pf}_4({\mathbf X})$. The module $\mathcal N$ is a self-dual maximal Cohen-Macaulay module of rank two; furthermore, $J$ is a Bourbaki ideal for $\mathcal N$. The ideals which define the homogeneous coordinate rings of the Pl\"ucker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module $\mathcal N$.