Automatic Uniformity
نویسنده
چکیده
Let X be an algebraic variety over the algebraically closed field K and Ξ ⊆ X(K) a set of K-rational points on X. We say that a subvariety Y ⊆ Xn of some Cartesian power of X is Ξ-special if Ξn ∩ Y (K) is Zariski dense in Y . We show under certain hypotheses on Ξ, for instance, that the class of Ξ-special varieties is closed under intersections, that when subvarieties of X vary in an algebraic family, the Zariski closures of their intersections with Ξ also vary uniformly. As a special case, we see that if the André-Oort conjecture holds, then for each g, n, and d there is a constant B = B(g, n, d) such that if C is a curve of degree d in the moduli space of g-dimensional principally polarized abelian varieties with full level n structure, either C is a modular curve or C contains at most B CM-moduli points (see Theorem 4.3).
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