An Elementary Approach to Containment Relations Between Symbolic and Ordinary Powers of Certain Monomial Ideals
arXiv:1512.07092
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 be a smooth projective variety of dimension , a non-zero sheaf of radical ideals with zero scheme ; if every irreducible component of has codimension at least , then \[ S^{(me)}_Z \subseteq S^m_Z \] for all . 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 . We work over an arbitrary field . Our main result is as follows. Let be a set of points not lying in a hyperplane. Then \[ S_Σ^{(\lceil (2-\frac{2}{n})m \rceil) } \subseteq S^m_Σ \] for all positive integers . If , then the same statement holds for three distinct points in arbitrary position.
5 pages