Fundamental Groups of Galois Closures of Generic Projections

Details Fundamental Groups of Galois Closures of Generic Projections Fundamental Groups of Galois Closures of Generic Projections

2005-05-19

arxiv.org/abs/math/0505406v2

Trans. AMS 362, 2167-2188 (2010)

Mathematics

Algebraic Geometry

18 pages, improved version

Scientific paper

For the Galois closure $\Xgal$ of a generic projection from a surface $X$, it is believed that $\pi_1(\Xgal)$ 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 $\pi_1(\Xgal)$ 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 $\pi_1(\Xgal)$. As a byproduct, we simplify part of the computations of Moishezon, Teicher and others.

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