Fabrizio Catanese: Local fundamental groups and surface classification (improving a theorem of Mumford)
 
 

Abstract.

In 1961 D. Mumford solved a conjecture by Abhyankar to the effect that given a point of an algebraic surface , it is analytically a smooth point iff it is topologically simple. More precisely, the triviality of the local fundamental group implies smoothness, which was proved by showing that if D is a normal crossings union of curves in a smooth surface S , and
i) the fundamental group of T-D ( T being a tubular neighbourhood of D)  is trivial
ii) the intersection matrix of D is negative definite
then
D is obtained by successive blow ups of a smooth point. Recent questions on fundamental groups had led me to extend Mumford's
theorem in the following way :

THM. D being as above , assume that K is nef on D . Then, if D_i is an irreducible component of D , and  \gamma_i is a small loop around D_i , we have that  \gamma_i  is non trivial.

To explain the title, recall that if K is not nef, either the surface is obtained blowing up a smooth point, or the surface is ruled.  Moreover, the method allows a   classification of the cases where \gamma_i is not of infinite order.