Covering relations between ball-quotient orbifolds

Abstract

Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinitely many orbifolds on P 2 uniformized by the complex 2-ball B 2 and some orbifolds over K3 surfaces uniformized by B 2 . We also give, along with infinitely many reducible examples, an infinite series of irreducible curves along which P 2 is uniformized by the product of 1-balls B 1 × B 1 .

Fundamental groups of some quadric-line arrangements

“…[9], 7.5), hence the meridians of irreducible components of C are supplementary invariants of the pair (P 2 , C). These meridians are specified in presentations of the fundamental group given below, they will be used in orbifoldfundamental group computations in [16]. Theorem 1.…”

“…Moreover, some line arrangements defined by unitary reflection groups studied in [12] are related to A 3 via orbifold coverings. For example, if L is the line arrangement z 2 ] ∈ P 2 is the arrangement A 3 , see [16] for details.…”

Galois Coverings of the Plane by K3 Surfaces

“…Note that any two such arrangements are projectively equivalent. This arrangement is named the Apollonius cycle by Holzapfel and Vladov [8] and shown to support several ball and bidisc-quotient orbifolds (see also [15]). …”

Fundamental Groups of Some Special Quadric Arrangements

“…Secondly, π 1 (P 2 \C) contributes to the study of the Galois coverings X → P 2 branched along C. Many interesting surfaces have been constructed as branched Galois coverings of the plane. One example is the arrangement A 3 (shown in Figure 1 below), which has Galois coverings X → P 2 branched along it; X ≃ P 1 × P 1 , or X is either an abelian surface, a K3 surface, or a quotient of the two-ball B 2 (see [9], [17], [19]). Moreover, some line arrangements defined by unitary reflection groups studied in [13] are related to A 3 via orbifold coverings.…”

“…then the image of L under the branched covering map [x : y : z] ∈ P 2 → [x 2 : y 2 : z 2 ] ∈ P 2 is the arrangement A 3 , see [17] for details. We use the algorithm of Moishezon-Teicher [12] in order to compute the braid monodromy of each one of the arrangements.…”