On the topology of some quasi-projective surfaces

Let $X$ be surface with isolated singularities in the complex projective space $P^3$ and let denote $Y$ the smooth part of $X$. In this note we discuss some aspects of the topology of such quasi-projective surfaces $Y$: the fundamental groups and the associated Galois coverings, the second homotopy groups and the mixed Hodge structure on the first cohomology group.Comment: version 2 contains new examples and many new references following suggestions by Ciro Ciliberto and De-Qi Zhan

Free and nearly free curves from conic pencils

We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.Comment: v3. Additional references adde