Abstract
The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.
Highlights
In his pioneering papers [4, 7], Hirzebruch constructed some new examples of algebraic surfaces which are ballquotients, i.e. algebraic surfaces for which the universal cover is the 2 -dimensional unit ball
The key idea of Hirzebruch, which enabled constructing new examples of ball-quotients, is that one can consider abelian covers of the complex projective plane branched along line arrangements [1]
Looking at Hirzebruch’s construction, one might hope that if suitably adapted, it can yield new examples of ball-quotients coming from certain curve arrangements in algebraic surfaces
Summary
In his pioneering papers [4, 7], Hirzebruch constructed some new examples of algebraic surfaces which are ballquotients, i.e. algebraic surfaces for which the universal cover is the 2 -dimensional unit ball. The key idea of Hirzebruch, which enabled constructing new examples of ball-quotients, is that one can consider abelian covers of the complex projective plane branched along line arrangements [1]. In the first part of the paper, we apply Hirzebruch’s construction to rational section arrangements in Hirzebruch surfaces and we show that it is not possible to construct new examples of ball-quotients. We present a general result which tells us that smooth algebraic surfaces W with KW nef and effective and certain curve arrangements coming from ample and effective linear systems do not provide new examples of ball-quotients.
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