Mathematics, Department of


Date of this Version



arXiv:1809.06955v5 [math.AC] 12 Nov 2020

doi 10.1016/j.jpaa.2020.106435


Copyright © 2020 Eloísa Grifo

Published: Journal of Pure and Applied Algebra Volume 224, Issue 12, December 2020, 106435


The symbolic powers I(n) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(n)Im holds. When I is an ideal of height 2 in a regular ring, I(3)I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (ta,tb,tc). More generally, given a radical ideal I of big height h, while the containment I(hnh+1)In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee such containments for n≫0.