Special apolar subset: the case of star configurations

TL;DR

This paper studies the existence of special point configurations called star configurations that are apolar to a generic degree d form in projective space, providing a complete characterization except for a specific case.

Contribution

It offers a comprehensive analysis of star configurations apolar to a generic form, including an algorithmic approach for unresolved cases.

Findings

Complete characterization of star configurations apolar to a generic form for most parameters.

An algorithmic method for the case (d, d+1, 2).

Advances understanding of apolarity and geometric configurations in algebraic geometry.

Abstract

In this paper we consider a generic degree $d$ form $ F $ in $n+1$ variables. In particular, we investigate the existence of star configurations apolar to $F$, that is the existence of apolar sets of points obtained by the $ n $-wise intersection of $ r $ general hyperplanes of $ \mathbb{P}^n $. We present a complete answer for all values of $(d,r,n)$ except for $(d,d+1,2)$ when we present an algorithmic approach.