On the Hilbert function on $\mathbb P^1\times\mathbb P^1$

TL;DR

This paper investigates how the Hilbert function of zero-dimensional schemes in the product of two projective lines changes when points are added along specific lines, providing new insights especially for ACM schemes.

Contribution

It characterizes the behavior of Hilbert functions of zero-dimensional schemes in $P^1 imes P^1$ when points are added on lines, including the ACM case.

Findings

Hilbert function changes when points are added on lines

Results hold under certain conditions for general schemes

For ACM schemes, the behavior is fully characterized

Abstract

Let $Q = \mathbb P^1 x \mathbb P^1$ and let $X\subset Q$ be a 0-dimensional scheme. This paper is a first step towards the characterization of Hilbert functions of 0- dimensional schemes in $Q$. In particular we show how, under some conditions on $X$, its Hilbert function changes when we add points to $X$ lying on a $(1,0)$ or $(0, 1)$-line. As a particular case we show also that if $X$ is ACM this result holds without any additional hypothesis.