  On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq
4$, we compute lower bounds for the {\em effective threshold} of an ample
divisor, in other words, the highest multiplicity a section of the divisor can
have at a specified point. We expect that these bounds are close to being
sharp. This translates into finding divisor classes on the blowup of
$\mathbb{P}(a,b,c)$ that generate a cone contained in, and probably close to,
the effective cone.
