Fundamental Groups of Galois Closures of Generic Projections

Abstract

For the Galois closure $\Xgal$ of a generic projection from a surface $X$, it is believed that $π_1(\Xgal)$ gives rise to new invariants of $X$. However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of $π_1(\Xgal)$ that depends on ${\pi }_1(X)$ and data from the generic projection only. In all known examples except one, this quotient is in fact isomorphic to $π_1(\Xgal)$. As a byproduct, we simplify part of the computations of Moishezon, Teicher and others.

18 pages, improved version

References in corpus (2)

• Higher degree Galois covers of CP^1 x T
• On Chisini's Conjecture. II

Cited by in corpus (2)

• Galois closure and Lagrangian varieties
• Natural Central Extensions of Groups