Abstract

There has recently been increased interest in understanding the relationship between the symbolic powers of an ideal and the geometric properties of the corresponding variety. While a number of results are available for the two-dimensional case, higher dimensions are largely unexplored. In the present paper we study a natural conjecture arising from a result by Bocci and Chiantini. As a first step toward understanding the higher-dimensional picture, we show that this conjecture is true in dimension three. Also, we provide examples showing that the hypotheses of the conjecture may not be weakened.

Dedicated to Robert Lazarsfeld on the occasion of his sixtieth birthday

1 Introduction

The study of the effect of points fattening was initiated by Bocci and Chiantini [3]. Roughly speaking, they considered the radical ideal I of a finite set Z of points in the projective plane, its second symbolic power I(2), and deduced from the comparison of algebraic invariants of these two ideals various geometric properties of the set Z. Along these lines, Dumnicki et al. [7] studied higher symbolic powers of I. Similar problems were studied in [1] in the bi-homogeneous setting of ideals defining finite sets of points in ℙ1 × ℙ1.

It is a natural task to try to generalize the result of Bocci and Chiantini [3, Theorem 1.1] to the higher-dimensional setting. Denoting by α (I) the initial degree of a homogeneous ideal I, i.e., the least degree k such that (I)k ≠ 0, a natural generalization reads as follows:

Conjecture 1.1Let Z be a finite set of points in projective space ℙn and let I be the radical ideal defining Z. If

d := α (I(n)) = α(I) + n − 1,

then either

α(I) = 1, i.e., Z is contained in a single hyperplane H in ℙn

or