Fundamental groups of Galois closures of generic projections

Liedtke, Christian

doi:10.1090/S0002-9947-09-04941-1

Fundamental groups of Galois closures of generic projections
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Trans. Amer. Math. Soc. 362 (2010), 2167-2188 Request permission

Abstract:

For the Galois closure $X_{\textrm {gal}}$ of a generic projection from a surface $X$, it is believed that $\pi _1(X_{\textrm {gal}})$ 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 $\pi _1(X_{\textrm {gal}})$ that depends on $\pi _1(X)$ and data from the generic projection only. In all known examples, this quotient is in fact isomorphic to $\pi _1(X_{\textrm {gal}})$. As a byproduct, we simplify the computations of Moishezon, Teicher and others.

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