Fundamental group of Log del Pezzo surfaces A Log del Pezzo surface is a normal complex surface with ample anti-canonical bundle and at worst quotient singularities. 
It is known that such surfaces are rational. This is proven, for example, here (Lemma 1.3):
http://www.ms.u-tokyo.ac.jp/journal/pdf/jms010104.pdf
I would like to know if there is some place with a short and clean exposition of this result, or maybe you can explain an idea of a proof? 
Note, that it is not hard to see that a Log del Pezzo has Kodaira dimension $-\infty$, so to prove that it is rational it is enough to prove that it has a trivial fundamental group.
 A: I would proceed as follows. Replacing $S$ by its minimal resolution I have a weak log Fano pair $(S,B)$ which is klt and $-(K_S+B)$ is semiample and big. In particular $-K_S\sim _Q -(K_S+B)+B$ is big and so $K_S$ is not pseudo-effective. Running the MMP $S\to S'$ I end up with either 1) a smooth del Pezzo surface of Picard number 1 (hence $\mathbb P ^2$) or 2) a Mori fiber space say $f:S'\to C$. We must then argue that $C$ is rational. There is probably a trivial way, but from the higher dimensional point of view, I would pick $H\sim _Q -(K_S+B)$ general so that $(S,B+H)$ is klt and then apply Kawamata's canonical bundle formula to $K_{S'}+B'+H'$ over $C$. Via some technical arguments, one can show that $K_{S'}+B'+H'\sim _Q f^*(K_C+G)$ where $G$ is big (we use the fact that $H'$ is big). But then as $\kappa (K_{S'}+B'+H')=0$, it follows that $\kappa (K_C+G)=0$ and hence $g(C)=0$.
One can generalize much of this argument to arbitrary dimension. In fact we can show that if $(X,B)$ is a klt log Fano then it is rationally connected see http://arxiv.org/pdf/math/0408301.pdf and http://arxiv.org/pdf/math/0504330v2.pdf (these proofs use Kawamata's canonical bundle formula: Y. Kawamata,
Subadjunction of log canonical divisors. II, Amer.J. Math.
120 (5)
(1998), 893-899). Of course, if $X$ is rationally connected and $X\to C$ is surjective, then $C$ is rationally connected, so if $\dim C=1$, then $C$ is a rational curve.
