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.

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    $\begingroup$ 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? $\endgroup$ – Mohan Aug 7 at 21:12
  • $\begingroup$ 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. $\endgroup$ – S. carmeli Aug 8 at 6:17

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