Hilbert functions and set of points in $${\varvec{\mathbb {P}}^{1}\times {\mathbb {P}}^{1}}$$ P 1 × P 1

Abstract

In this paper we study the problem of classifying the Hilbert functions of zero-dimensional schemes in \(\mathbb P^1\times \mathbb P^1\). In particular, in the main result of the paper we give conditions to determine some Hilbert functions of set of points in \(\mathbb P^1\times \mathbb P^1\) and we describe geometrically these schemes. Moreover, we show that the Hilbert functions of these schemes depend only on the distribution of the points on a set of \((1,0)\) and \((0,1)\)-lines.