We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...

We deal with a reducible projective surface $X$ with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the $\omega$-genus$p_{\omega }\left(X\right)$ of $X$, i.e. the dimension of the vector space of global sections of the dualizing sheaf ${\omega }_X$. Then we prove that, when $X$ is smoothable, i.e. when $X$ is the central fibre of a flat family $\pi :𝒳\to \Delta$ parametrized by a disc, with smooth general fibre, then the $\omega$-genus of the fibres of $\pi$ is constant.