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On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq4$, we compute lower bounds for the {\em effective threshold} of an ampledivisor, in other words, the highest multiplicity a section of the divisor canhave at a specified point. We expect that these bounds are close to beingsharp. 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.

On curves with high multiplicity on $\mathbb{P}(a,b,c)$ for $\min(a,b,c)\leq4$