Abstract. We study effective divisors $D$ on surfaces with $H^0(\mathcal{O}_D)=k$ and $H^1(\mathcal{O}_D)=H^0(\mathcal{O}_D(D))=0$. We give a numerical criterion for such divisors, following a general investigation of negativity, rigidity and connectivity properties. Examples include exceptional loci of rational singularities, and spherelike divisors.