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.

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with pg = 0 and K2 = 6. We also give a new description for these surfaces, extending ideas of Inoue. Finally, we calculate the bicanonical degree of Kulikov surfaces and prove that they verify the Bloch conjecture.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by $\text{L}$. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

For a variety X of dimension n in ${\mathbb P}^r,\ r\geq n(k+1)+k$, the kth secant order of X is the number $\mu_k(X)$ of $(k+1)$-secant k-spaces passing through a general point of the kth secant variety. We show that, if $r>n(k+1)+k$, then $\mu_k(X)=1$ unless X is k-weakly defective. Furthermore we give a complete classification of surfaces $X\subset{\mathbb P}^r,\ r>3k+2$, for which $\mu_k(X)>1$.

We describe the irrational pencils on surfaces of general type with ${{p}_{g}}=q=2$.

In this paper we prove that a normal Gorenstein surface dominated by P2 is isomorphic to a quotient P2/G, where G is a finite group of automorphisms of P2 (except possibly for one surface ). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.