Galois Coverings of the Plane by K3 Surfaces

Abstract

Abstract. We study branched Galois coverings of the projective plane by smooth K3 surfaces. Branching data of such a covering determines in a unique way a uniformizable orbifold on the plane. In order to study Galois coverings of the plane by K3 surfaces, it suffices to study orbifolds on the plane uniformized by K3 surfaces. We call these K3 orbifolds and classify K3 orbifolds with an abelian uniformization. We also classify K3 orbifolds with a locus of degree less than 6 and with a non-abelian uniformization. There are no K3 orbifolds with a locus of degree greater than 6. Although we give some examples of K3 orbifolds with a sextic locus, our results are incomplete in this case.

Fundamental Groups of a Class of Rational Cuspidal Plane Curves

“…. , n, and we arrive at the Hirzebruch surface X n+1 with E 2 n+1 = −n − 1 (see Figure (12)). Performing elementary transformations at arbitrary points s i ∈ P i \E i for i = n + 1, .…”

“…). (See Figure (12), where the situation is illustrated for n = 2, m = 1, beware that the elementary transformation applied at e := E 1 ∩ P 1 and the elementary transformation applied at ẽ := E 2 ∩ Q 2 are shown simultaneously in the figure .) Recall that the subsequent transformations are applied at points…”

“…The idea is an analogue of the leading principle in the knot theory: in order to understand a knot K ⊂ S 3 , look at the topology of the knot complement S 3 \K. Another strong motivation for studying π 1 (P 2 \C) comes from the surface theory: A good knowledge of π 1 (P 2 \C) allows one to construct Galois coverings of P 2 branched at C (see [12] for explicit examples). Moreover, if X → P 2 is a branched covering, with C as the branching locus, one can hope to derive some invariants of X from the invariants of the topology of P 2 \C.…”

Smooth finite abelian uniformizations of projective spaces and Calabi–Yau orbifolds

“…Their quotients sometimes yields CY orbifolds with a non-abelian uniformizing group. In dimension 2, many examples were constructed in [5]. In dimension 3 consider the CY orbifold [2, 2, 2, 2, 2, 2, 2, 2].…”

“…In dimension one, this classification is classical. K3 orbifolds with a locus of degree ≤ 5 were classified in [5]. There are no K3 orbifolds with a locus of degree > 6.…”