An Elementary Approach to Containment Relations Between Symbolic and Ordinary Powers of Certain Monomial Ideals

TL;DR

This paper provides an elementary proof of a known containment relation between symbolic and ordinary powers of ideals, focusing on sets of points in projective space and avoiding advanced algebraic geometry techniques.

Contribution

It offers a simplified, elementary proof of a key containment theorem for symbolic and ordinary powers of ideals, specifically for points in projective space, reducing the problem to coordinate axes ideals.

Findings

Proves a specific containment relation for sets of points in projective space.

Reduces the general case to ideals of coordinate axes in affine space.

Provides results for n points not in a hyperplane and for three points in arbitrary position.

Abstract

The purpose of this note is to find an elemenary explanation of a surprising result of Ein--Lazarsfeld--Smith \cite{ELS} and Hochster--Huneke \cite{HH} on the containment between symbolic and ordinary powers of ideals in simple cases. This line of research has been very active ever since, see for instance \cites{BC,HaH,DST} and the references therein, by now the literature on this topic is quite extensive. By `elementary' we refer to arguments that among others do not make use of resolution of singularities and multiplier ideals nor tight closure methods. Let us quickly recall the statement \cite{ELS}: let $X$ be a smooth projective variety of dimension $n$, $S \subseteq O_X$ a non-zero sheaf of radical ideals with zero scheme $Z\subseteq X$; if every irreducible component of $Z$ has codimension at least $e$, then \[ S^{(me)}_Z \subseteq S^m_Z \] for all $m\geq 1$. Our goal is to reprove this assertion in the case of points in projective spaces (as asked in \cite{PAGII}*{Example 11.3.5}) without recurring to deep methods of algebraic geometry. Instead of working with subsets of projective space, we will concentrate on the affine cones over them; our aim hence becomes to understand symbolic and ordinary powers ideals of sets of line through the origin. We will end up reducing the general case to a study of the ideals \[ I_{2,n} =(x_ix_j\mid 1\leq i<j\leq n) \subseteq k[x_1,\dots,x_n] \] defining the union of coordinate axes in $\AA^n_{k}$. We work over an arbitrary field $k$. Our main result is as follows. Let $\Sigma\subseteq P^{n-1}$ be a set of $n$ points not lying in a hyperplane. Then \[ S_{\Sigma}^{(\lceil (2-\frac{2}{n})m \rceil) } \subseteq S^m_{\Sigma} \] for all positive integers $m$. If $n=3$, then the same statement holds for three distinct points in arbitrary position.