Computing Negative Curves on Blowups of Weighted Projective Planes

We present a computational method for detecting negative curves of prescribed deficiency on the blowup of a weighted projective plane at a general smooth point. The method combines explicit bounds from the orbifold Riemann–Roch formula with a finite linear-algebra test on weighted homogeneous polynomials, yielding verifiable certificates for the curves found. For pairwise coprime weights 0<a<b<c≤100 satisfying (a+b+c)2<abc, we classify the existence of deficiency-0 negative curves and determine the exceptional triples with none. Among these exceptional cases, we further classify deficiency-1 negative curves, recovering the previously known examples and producing two new ones, and we prove the nonexistence of deficiency 2 and 3 negative curves in the remaining cases in the same range.