The order of finite algebraic fundamental groups of surfaces with K^2<=3\chi-2
We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group of S has order at most 5, and equality only occurs if S is a Godeaux surface. We also show that if K_S^2<= 3\chi-3 and the algebraic fundamental group of S is not trivial, then it is Z_2, or Z_2^2 or Z_3. Furthermore in this last case one has: 2<=\chi<=4, K^2=3\chi-3 and these possibilities do occur.
NUMERICAL CAMPEDELLI SURFACES WITH FUNDAMENTAL GROUP OF ORDER 9
We give explicit constructions of all the numerical Cam\-pe\-delli surfaces, i.e.\ the minimal surfaces of general type with $K^2=2$ and $p_g=0$, whose fundamental group has order 9. There are three families, one with $\pionealg=\Z_9$ and two with $\pionealg=\Z_3^2$. We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with $\pionealg=\Z_9$ and for one of the families of surfaces with $\pionealg=\Z_3^2$ the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces of general type with $K^2>1$ whose bicanonical system has base points.
A simply connected numerical Godeaux surface with ample canonical class
We prove that a recent construction of a numerical Godeaux surface due to P. Craighero and R. Gattazzo is simply connected, and show how to realize their construction as a double plane. By proving that the surface contains no (-2)-curves, we obtain that this is the first example of a simply connected surface with vanishing geometric genus and ample canonical class.