Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O\_S)=0$ and $P\_2(S)=h^0(\mathcal O\_S(2K\_S))=0$. If $S$ is smooth, Castelnuovo's rationality criterion (proved by Zariski in characteristic p) tells us that $S$ is rational. 


Does this extend to singular surfaces if the characteristic is 0? I'm happy if it extends only to 'some' singular surfaces (e.g. canonical).


I have looked for this in the literature but I haven't found anything to prove it or disprove it. I assume someone has asked this question before (although not in MO, it seems).



Castelnuovo's rationality criterion on singular surfaces?
It does not hold in general: a cone over a smooth plane cubic satisfies $q=P\_2=0$ but is not rational.


On the other hand if $S$ has canonical singularities and $\tilde{S} \rightarrow S$ is any resolution, one has $P\_2(S)=P\_2(\tilde{S} )$ and $q(S)=q(\tilde{S} )$, hence $\tilde{S} $ and therefore $S$ are rational.


