  We study the birational geometry of Hilbert schemes of points on non-minimal
surfaces. In particular, we study the weak Lefschetz Principle in the context
of birational geometry. We focus on the interaction of the stable base locus
decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and
$Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between
surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask
if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$
yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if the weak
Lefschetz Principle holds. Even though the stable base loci in $N^1(X^{[n]})$
fails to provide information about how the two decompositions interact, we show
that the restriction of the augmented stable base loci of $X^{[n]}$ to
$Y^{[n]}$ is equal to the stable base locus decomposition of $Y^{[n]}$. We also
exhibit effective divisors induced by Severi varieties. We compute the classes
of such divisors and observe that in the case that $X$ is the projective plane,
these divisors yield walls of the SBLD for some cases.
