2013/03/09 by Ingrid Bauer, Fabrizio Catanese, Bauer, Ingrid +3 · 1 citation
Mathematics · #11R32 #14 M99 #14J10 #14J29 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG #msc:11R32 #msc:14 #msc:14J10 #msc:14J29 #msc:M99
paper · pdf · doi:10.48550/arxiv.1303.2248
24 pages, extends and corrects a previous article arXiv:0706.1466
arxiv created 2013/03/09 · openalex publication_date 2013/03/09 · arxiv updated 2013/03/12 · openalex created_date 2022/08/13 · openalex updated_date 2026/07/28
We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve Ca, and then a triangle curve (Da, Ga) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal(\Q /\Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C1 x C2 of curves of respective genera g1, g2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element σ∈ \Gal(\Q /\Q), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface Xσ have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.