In this paper, we construct the first examples of complex surfaces of general type with arbitrarily large geometric genus whose canonical maps induce non-hyperelliptic fibrations of genus $g=4$, and on the other hand, we prove that there is no complex surface of general type whose canonical map induces a hyperelliptic fibrations of genus $g\geqslant 4$ if the geometric genus is large.

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod $p$ technique and deformation theory.

Let x0, x1, x2, x3 be polynomials in a variable t and with coefficients in a field k of character of characteristic 0. If and , then x0 = x1 = x2 = x3 = 0. This partially answers a question of Pjatetskii-Š;apiro and Šafarevič about the K3-surface . The proof uses a technique of M. R. Christie.