On the Hilbert function of zero-dimensional schemes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb P}^1 \times {\mathbb P}^1}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q = {\mathbb P}^1 \times {\mathbb P}^1}$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subset Q}$$\end{document} be a zero-dimensional scheme. The results in this paper give the possibility of computing, under certain hypotheses, the Hilbert function of a zero-dimensional scheme in Q that is not ACM. 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.