Surface classification and local and global fundamental groups, I

2006/02/07 by Catanese, Fabrizio · 1 citation
#14B05 #14J80 #20F05 #32S50 #57M05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.math/0602128

Abstract

Given a smooth complex surface S, and a compact connected global normal crossings divisor D = ∪_i D_i, we consider the local fundamental group, i.e., the fundamental group Gamma of T-D, where T is a good tubular neighbourhood of D. One has a surjection of Gamma onto the fundamental group of D, and the kernel \sK is normally generated by geometric loops \ga_i around the curve D_i. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of \ga_i in the local fundamental group, provided all the curves D_i of genus zero have selfintersection <= -2. (in particular this holds if the canonical divisor is nef on D), and under the technical assumption that the dual graph of D is a tree.

Cited by

• A new coding theory, for normal surfaces, and ADE singularities, I

Related

• New Invariants for surfaces
• Infinite presentations for fundamental groups of surfaces
• Lectures on Klein surfaces and their fundamental group
• Fundamental groups of complements of curves as solvable groups
• A representation theoretic characterization of simple closed curves on a surface