#### Vol. 12, No. 5, 2018

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Characterization of Kollár surfaces

### Giancarlo Urzúa and José Ignacio Yáñez

Vol. 12 (2018), No. 5, 1073–1105
##### Abstract

Kollár (2008) introduced the surfaces

$\left({x}_{1}^{{a}_{1}}{x}_{2}+{x}_{2}^{{a}_{2}}{x}_{3}+{x}_{3}^{{a}_{3}}{x}_{4}+{x}_{4}^{{a}_{4}}{x}_{1}=0\right)\subset ℙ\left({w}_{1},{w}_{2},{w}_{3},{w}_{4}\right)$

where ${w}_{i}={W}_{i}∕{w}^{\ast }$, ${W}_{i}={a}_{i+1}{a}_{i+2}{a}_{i+3}-{a}_{i+2}{a}_{i+3}+{a}_{i+3}-1$, and ${w}^{\ast }=gcd\left({W}_{1},\dots ,{W}_{4}\right)$. The aim was to give many interesting examples of $ℚ$-homology projective planes. They occur when ${w}^{\ast }=1$. For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For ${w}^{\ast }>1$, we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers ${z}^{{w}^{\ast }}={l}_{1}^{{a}_{2}{a}_{3}{a}_{4}}{l}_{2}^{-{a}_{3}{a}_{4}}{l}_{3}^{{a}_{4}}{l}_{4}^{-1}$, where $\left\{{l}_{1},{l}_{2},{l}_{3},{l}_{4}\right\}$ are four general lines in ${ℙ}^{2}$. In addition, by using various properties on classical Dedekind sums, we prove that:

1. For any ${w}^{\ast }>1$, we have ${p}_{g}=0$ if and only if the Kollár surface is rational. This happens when ${a}_{i+1}\equiv 1$ or ${a}_{i}{a}_{i+1}\equiv -1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}{w}^{\ast }\right)$ for some $i$.
2. For any ${w}^{\ast }>1$, we have ${p}_{g}=1$ if and only if the Kollár surface is birational to a K3 surface. We classify this situation.
3. For ${w}^{\ast }\gg 0$, we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with ${K}_{S}^{2}∕e\left(S\right)\to 1$.

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$\mathbb Q$-homology projective planes, Dedekind sums, branched covers