The Aluffi algebra of the Jacobian of points in projective space: torsion-freeness

TL;DR

This paper investigates the properties of the Aluffi algebra associated with the Jacobian ideal of points in general linear position in projective space, focusing on torsion-freeness and algebraic structure.

Contribution

It characterizes when the Aluffi algebra of the Jacobian ideal of points in general position is torsion-free, providing new insights into its algebraic properties.

Findings

Identification of conditions for torsion-freeness of the Aluffi algebra

Analysis of the algebraic structure of Jacobian ideals of points in projective space

Establishment of new links between Aluffi and Rees algebras in this context

Abstract

The algebra in the title has been introduced by P. Aluffi. Let $J\subset I$ be ideals in the commutative ring $R$. The (embedded) Aluffi algebra of $I$ on $R/J$ is an intermediate graded algebra between the symmetric algebra and Rees Algebra of the ideal $I/J$ over $R/J$. A pair of ideals has been dubbed an Aluffi torsion-free pair if the surjective map of the Aluffi algebra of $I/J$ onto the Rees algebra of $I/J$ is injective. In this paper we focus on the situation where $J$ is the ideal of points in general linear position in projective space and $I$ is its Jacobian ideal.