Characterization of Koll\'ar surfaces

Koll\'ar introduced in [Ko08] the surfaces $$(x_1^{a_1}x_2+x_2^{a_2}x_3+x_3^{a_3}x_4+x_4^{a_4}x_1=0)\subset \mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^*$, $W_i=a_{i+1}a_{i+2}a_{i+3}-a_{i+2}a_{i+3}+a_{i+3}-1$, and $w^*=$gcd$(W_1,\ldots,W_4)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^*=1$. For that case, we prove that Koll\'ar surfaces are Hwang-Keum [HK12] surfaces. For $w^*>1$, we construct a geometrically explicit birational map between Koll\'ar surfaces and cyclic covers $z^{w^*}=l_1^{a_2 a_3 a_4} l_2^{-a_3 a_4} l_3^{a_4} l_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in $\mathbb{P}^2$. In addition, by using various properties on classical Dedekind sums, we prove that: (a) For any $w^*>1$, we have $p_g=0$ iff the Koll\'ar surface is rational. This happens when $a_{i+1} \equiv 1$ or $a_{i}a_{i+1} \equiv -1 ($mod $w^*)$ for some $i$. (b) For any $w^*>1$, we have $p_g=1$ iff the Koll\'ar surface is birational to a K3 surface. We classify this situation. (c) For $w^*>>0$, we have that the smooth minimal model $S$ of a generic Koll\'ar surface is of general type with $K_{S}^2/e(S) \to 1$.