## Algebra & Number Theory

### Characterization of Kollár surfaces

#### Abstract

Kollár (2008) introduced the surfaces

$( x 1 a 1 x 2 + x 2 a 2 x 3 + x 3 a 3 x 4 + x 4 a 4 x 1 = 0 ) ⊂ P ( w 1 , w 2 , w 3 , w 4 )$

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 $w ∗ = gcd ( W 1 , … , W 4 )$. The aim was to give many interesting examples of $ℚ$-homology projective planes. They occur when $w ∗ = 1$. For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For $w ∗ > 1$, we construct a geometrically explicit birational map between Kollár 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 $P 2$. In addition, by using various properties on classical Dedekind sums, we prove that:

1. For any $w ∗ > 1$, we have $p g = 0$ if and only if the Kollár surface is rational. This happens when $a i + 1 ≡ 1$ or $a i a i + 1 ≡ − 1 ( mod w ∗ )$ for some $i$.
2. For any $w ∗ > 1$, we have $p g = 1$ if and only if the Kollár surface is birational to a K3 surface. We classify this situation.
3. For $w ∗ ≫ 0$, we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with $K S 2 ∕ e ( S ) → 1$.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 5 (2018), 1073-1105.

Dates
Received: 23 December 2016
Revised: 29 January 2018
Accepted: 17 March 2018
First available in Project Euclid: 14 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1534212097

Digital Object Identifier
doi:10.2140/ant.2018.12.1073

Mathematical Reviews number (MathSciNet)
MR3840871

Zentralblatt MATH identifier
06921170

Subjects
Primary: 14J10: Families, moduli, classification: algebraic theory

#### Citation

Urzúa, Giancarlo; Yáñez, José Ignacio. Characterization of Kollár surfaces. Algebra Number Theory 12 (2018), no. 5, 1073--1105. doi:10.2140/ant.2018.12.1073. https://projecteuclid.org/euclid.ant/1534212097