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. 
Andreas Hochenegger and David Ploog
math.AG