# Resolution of rational surfaces

Let $$S$$ be a rational singular complete algebraic surface over $$\mathbb{C}$$. Let $$\phi:\tilde{S}\to S$$ be a resolution of singularities with minimal possible Picard rank (i.e. minimal $$\mathrm{dim}(H_2(\tilde{S};\mathbb{Q}))$$). Is there a known a-priori bound on this rank in terms of some computable invariants of $$S$$? More generally, a bound in terms of invariants of some smooth deformation $$S'$$ of $$S$$ is fine as well, as long as it can be computed reasonably.

• If you take the rational surface given by $z^n-xy=0$, then the minimal resolution has Picard rank $n-1$. Can you tell me what computable invariant will tell me this? – Mohan Aug 7 at 21:12
• By a surface I mean a complete surface, ill edit to make it clear. In the case of hypersurface on projective space, my original hope was that, for example, the dimension of $H^{1,1}$ of a smooth surface of the same degree would work. – S. carmeli Aug 8 at 6:17