We relate properties of linear systems on
X
X
to the question of when
I
r
I^r
contains
I
(
m
)
I^{(m)}
in the case that
I
I
is the homogeneous ideal of a finite set of distinct points
p
1
,
…
,
p
n
∈
P
2
p_1,\ldots ,p_n\in \mathbf {P}^2
, where
X
X
is the surface obtained by blowing up the points. We obtain complete answers for when
I
r
I^r
contains
I
(
m
)
I^{(m)}
when the points
p
i
p_i
lie on a smooth conic or when the points are general and
n
≤
9
n\le 9
.