Rational Normal Curves and Hadamard Products

Abstract

Given $$r>n$$ r > n general hyperplanes in $$\mathbb P^n,$$ P n , a star configuration of points is the set of all the n-wise intersection of the hyperplanes. We introduce 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 $$\mathbb P^2$$ 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.

Hadamard products of symbolic powers and Hadamard fat grids

“…V X = {(2, 21), (5,17), (8,13), (11,9), (14,5), (17, 2)} and C X = {(0, 21), (2,17), (5,13), (8,9), (11,5), (14, 2), (17, 0)}.…”

“…This fact leads to the question if other interesting set of points can be constructed by Hadamard product. Recent results in this direction can be found in [3], where the second author along with C. Capresi and D. Carrucoli built Gorenstein set of points in P 3 , with given h−vector, by creating, via Hadamard products, a stick figure of lines to which they apply the results of Migliore and Nagel [21], and also in [9] where the third author along with E. Carlini, M.V. Catalisano and G. Favacchio found a relation between star configurations where all the hyperplanes are osculating to the same rational normal curve (contact star configurations) and Hadamard products of linear varieties.…”