Special apolar subset: the case of star configurations

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.