Grassmann cactus variety and socle dimension

TL;DR

This paper studies Grassmann cactus varieties, showing they can be characterized using finite schemes with low socle dimension, which simplifies their analysis and computation.

Contribution

It introduces a new characterization of Grassmann cactus varieties via schemes with low socle dimension, aiding in their study and calculation.

Findings

Characterization of Grassmann cactus varieties using schemes with low socle dimension

Simplification of calculations for Grassmann cactus varieties

Development of parameter spaces for schemes with low socle dimension

Abstract

Grassmann cactus variety is a common generalisation of Grassmann secant variety and cactus variety. In their definitions one considers the vector spaces of fixed dimension that are contained in the linear span of some finite schemes. We prove that to characterise Grassmann cactus varieties it is enough to use finite schemes that locally have low socle dimension. This motivates the study of parameter spaces of such schemes and simplifies calculations of examples of Grassmann cactus varieties.