Abstract view
Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$


Published:20121113
Printed: Aug 2013
Elena Guardo,
Dipartimento di Matematica e Informatica, Viale A. Doria, 695100Catania, Italy
Brian Harbourne,
Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 685880130, USA
Adam Van Tuyl,
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1
Abstract
Recent work of EinLazarsfeldSmith and HochsterHuneke
raised the problem of which symbolic powers of an ideal
are contained in a given ordinary power of the ideal.
BocciHarbourne developed methods to address this problem,
which involve asymptotic numerical characters of
symbolic powers of the ideals. Most of the work
done up to now has been done for ideals defining 0dimensional
subschemes of projective space.
Here we focus on certain subschemes given by
a union of lines in $\mathbb{P}^3$ which can also be viewed
as points in $\mathbb{P}^1 \times \mathbb{P}^1$.
We also obtain results on the
closely related problem, studied by Hochster and by LiSwanson, of
determining situations for which
each symbolic power of an ideal is an ordinary power.