Rigid divisors on surfaces

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.