Characterization of Kollár surfaces

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 $w^{*} = gcd (W_{1}, \dots, W_{4})$ . 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:

For any $w^{*} > 1$ , we have $p_{g} = 0$ if and only if the Kollár surface is rational. This happens when $a_{i + 1} \equiv 1$ or $a_{i} a_{i + 1} \equiv - 1 (mod w^{*})$ for some $i$ .
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.
For $w^{*} ≫ 0$ , we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with $K_{S}^{2} ∕ e (S) \to 1$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 14J10

Milestones

Received: 23 December 2016

Revised: 29 January 2018

Accepted: 17 March 2018

Published: 31 July 2018

Authors

Giancarlo Urzúa
Facultad de Matemáticas Pontificia Universidad Católica de Chile Campus San Joaquín Santiago Chile

José Ignacio Yáñez
Department of Mathematics University of Utah Salt Lake City, UT United States

Vol. 12, No. 5, 2018

Giancarlo Urzúa and José Ignacio Yáñez

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors