Rational normal curves and Hadamard products

TL;DR

This paper explores the relationship between contact star configurations, rational normal curves, and Hadamard products, revealing new geometric properties and conditions under which unions form complete intersections.

Contribution

It introduces contact star configurations linked to rational normal curves and Hadamard products, and analyzes their properties and unions in projective spaces.

Findings

Contact star configurations relate to Hadamard products of linear varieties.

The union of two contact star configurations on a conic has a special h-vector.

Under certain conditions, these unions form complete intersections.

Abstract

Given $r>n$ general hyperplanes in $\mathbb P^n,$ a star configuration of points is the set of all the $n$-wise intersection of them. We introduce {\it contact star configurations}, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in $\mathbb P^2$, we prove that the union of two contact star configurations has a special $h$-vector and, in some cases, this is a complete intersection.