# The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

## TL;DR

This paper investigates the Valabrega-Valla module of the Jacobian ideal for points in the projective plane, providing classifications for configurations of 5 and 6 points and identifying cases where the module is nonzero.

## Contribution

It offers a complete classification of the Valabrega-Valla module for 5 and 6 points in the projective plane, highlighting cases with non-vanishing modules for specific point configurations.

## Key findings

- The module is nonzero for certain special configurations.
- Complete classification for 5 and 6 points.
- Identification of configurations with vanishing modules.

## Abstract

The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.