On The Relation Between The Hilbert Functions of Fat Point Schemes

Abstract

Let $\phi: \mathbb P^{n-1} \to \mathbb P^n$ be the embedding, and $m_1P_1+\cdots+m_sP_s$ be a fat point scheme of $\mathbb P^{n-1}$. We show the relation between the Hilbert function of the fat point scheme $m_1P_1+\cdots+m_sP_s$ of $\mathbb P^{n-1}$ and the Hilbert function of the fat point scheme $m_1\phi(P_1)+\cdots+m_s\phi(P_s)$ of $\mathbb P^{n}$. Then we show an application of this relation.