  Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of
which symbolic powers of an ideal are contained in a given ordinary power of
the ideal. Bocci-Harbourne 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 0-dimensional
subschemes of projective space. Here we focus on certain subschemes given by a
union of lines in ${\bf P}^3$ which can also be viewed as points in ${\bf
P}^1\times {\bf P}^1$. We also obtain results on the closely related problem,
studied by Hochster and by Li-Swanson, of determining situations for which each
symbolic power of an ideal is an ordinary power.
