Abstract
A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.
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References
Belyi G V, Galois extensions of a maximal cyclotomic field,Izv. Akad. Nauk SSSR Ser. Mat. 43 (2) (1979) 267–276, 479
Zariski O, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves,Am. J. Math. 87 (1965) 507–536
Zariski O, Studies in equisingularity. II. Equisingularity in codimension 1 (and characteristic zero),Am. J. Math. 87 (1965) 972–1006
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Paranjape, K.H. A geometric characterization of arithmetic varieties. Proc. Indian Acad. Sci. (Math. Sci.) 112, 383–391 (2002). https://doi.org/10.1007/BF02829791
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DOI: https://doi.org/10.1007/BF02829791