The effect of points fattening in dimension three

TL;DR

This paper investigates a conjecture related to symbolic powers of ideals in three-dimensional algebraic varieties, confirming its validity in dimension three and providing counterexamples to hypothesis weakening.

Contribution

It proves a conjecture about points fattening in three dimensions and explores the limits of its hypotheses, advancing understanding in higher-dimensional algebraic geometry.

Findings

Conjecture holds in dimension three

Counterexamples show hypotheses cannot be weakened

Advances understanding of symbolic powers in higher dimensions

Abstract

There has been increased recent 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, the higher-dimensional case is largely unexplored. In the present paper we study a natural conjecture arising from a result by Bocci and Chiantini. As a first step towards 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.