Characterization of Koll\'ar surfaces

TL;DR

This paper studies Kollár surfaces, classifying their geometric properties, constructing explicit birational maps for certain cases, and analyzing their invariants using Dedekind sums, revealing when they are rational, K3, or of general type.

Contribution

It provides a detailed classification of Kollár surfaces, constructs explicit birational maps for cases with nontrivial gcd, and links their properties to Dedekind sums and classical surface theory.

Findings

Kollár surfaces are Hwang-Keum surfaces when w^*=1.

For w^*>1, a birational map relates Kollár surfaces to cyclic covers of P^2.

p_g=0 iff the surface is rational; p_g=1 iff it is a K3 surface.

Abstract

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 $K_{S}^2/e(S) \to 1$.