Rend. Circ. Mat. Palermo, II. Ser (2017) 66:233–245
On the containment problem
· J. Szpond
Received: 7 January 2016 / Accepted: 22 October 2016 / Published online: 28 October 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract The purpose of this note is to provide an overview of the containment problem
for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples.
We focus here on ideals with zero dimensional support. This is an area of ongoing active
research. We conclude the note with a list of potential promising paths of further research.
Keywords Symbolic power · Fat points · Waldschmidt constants
Mathematics Subject Classiﬁcation 14C20 · 14J26 · 14N20 · 13A15 · 13F20
Let I ⊂ R =
] be a homogeneous ideal. Such an ideal determines various
sequences of associated ideals. The most natural one is that of ordinary powers of I:
= R ⊃ I
= I ⊃ I
If I =f
is given explicitly in terms of generators, then the r-th ordinary power
of I is generated by products
Polish Academy of Sciences, Institute of Mathematics, ul.
Sniadeckich 8, 00-656 Warsaw, Poland
Department of Mathematics, Pedagogical University of Cracow, Podchor¸a˙zych 2, 30-084 Kraków,