On Nori's Fundamental Group Scheme

We determine the quotient category which is the representation category of the kernel of the homomorphism from Nori's fundamental group scheme to its \'etale and local parts. Pierre Deligne pointed out an error in the first version of this article. We profoundly thank him, in particular for sending us his enlightning example reproduced in Remark 2.4 2).Comment: 29 page

Dressing preserving the fundamental group

In this note we consider the relationship between the dressing action and the holonomy representation in the context of constant mean curvature surfaces. We characterize dressing elements that preserve the topology of a surface and discuss dressing by simple factors as a means of adding bubbles to a class of non finite type cylinders.Comment: 36 pages, 1 figur

The fundamental group and torsion group of Beauville surfaces

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.Comment: 14 pages; MAGMA script included; v2: minor corrections, final version to appear in the Proceedings of the Conference "Beauville Surfaces and Groups", Newcastle University (UK), 7-9th June 201

On non Fundamental Group Equivalent Surfaces

In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP^2 and a degenerations of the surface into a union of planes - the "pillow" degeneration for the non-prime surface and the "magician" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a projective deformation of each othe