Geometric properties derived from generic initial spaces

TL;DR

This paper explores the geometric properties of ideals generated by homogeneous forms through the structure of their generic initial spaces, extending previous results in algebraic geometry.

Contribution

It generalizes earlier findings by analyzing the generic initial ideal's form for vector spaces of homogeneous forms.

Findings

Characterization of generic initial spaces for homogeneous forms

Extension of previous algebraic geometry results

Deeper understanding of ideal structures in polynomial rings

Abstract

For a vector space V of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by V, from the form of the generic initial space gin(V) for the revlex order. Our main result is a considerable generalisation of a previous result by the first author.