Noncritical Belyi Maps

×ØÖÖغ In the present paper, we present a slightly strengthened version of a well-known theorem of Belyi on the existence of " Belyi maps ". Roughly speaking, this strengthened version asserts that there exist Belyi maps which are unramified at [cf. Theorem 2.5] — or even near [cf. Corollary 3.2] — a prescribed finite set of points. Write C for the complex number field; Q ⊆ C for the subfield of algebraic numbers. Let X be a smooth, proper, connected algebraic curve over Q. If F is a field, then we shall denote by P 1 F the projective line over F. Definition 1.1. We shall refer to a dominant morphism [of Q-schemes] 1 Q as a Belyi map if φ is unramified over the open subscheme U P ⊆ P 1 Q given by the complement of the points " 0 " , " 1 " , and " ∞ " of P 1 Q ; in this case, we shall refer to U X def = φ −1 (U P) ⊆ X as a Belyi open of X. In [1], it is shown that X always admits at least one Belyi open. From this point of view, the main result (Theorem 2.5) of the present paper has as an immediate formal consequence (pointed out to the author by A. Tamagawa) the following interesting [and representative] result: Corollary 1.2. (Belyi Opens as a Zariski Base) If V X ⊆ X is any open subscheme of X containing a closed point x ∈ X, then there exists a Belyi open U X ⊆ V X ⊆ X such that x ∈ U X. In particular, the Belyi opens of X form a base for the Zariski topology of X.