On a weighted projective surface P(a,b,c) with min(a,b,c) ≤ 4, we compute lower bounds for the 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 P(a,b,c) that generate a cone contained in, and probably close to, the effective cone.

The first author and third authors were partially supported by a Discovery Grant from the National Science and Engineering Board of Canada. The second and fourth authors were supported by an Undergraduate Student Research Award from the National Science and Engineering Board of Canada.