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Remarks about uniform boundedness of rational points over function fields. (English) Zbl 1030.14022

Let \(g\geq 2, d\geq 1, s\geq 0\) be fixed integers. Let \(V\) be a smooth, irreducible, projective variety of degree \(d\) defined over \( {\mathbb C}\). Let \(T \subset V\) be a closed subscheme of degree \(s\). The author introduces the set \(F_g(V,T)\) of equivalence classes of non-isotrivial families of smooth curves of genus \(g\) over \(V \setminus T\) and proves (theorem 1) that there exists a number \(H(g,d,s)\) such that \(|F_g(V,T)|\leq H(g,d,s)\). Moreover, the bound \(H\) does not depend on \(s\), if \(T\) has codimension at least 2 in \(V\). With the above notation, let \(X\) be a non-isotrivial curve of genus \(g\) defined over \(L= {\mathbb C}(V)\) and having good reduction outside \(T\). It is proven (theorem 2) that \(|X(L)|\leq N(g,d,s)\) for some integer \(N(g,d,s)\). Let \(C_g^2(L)\) be the set of \(L\)-isomorphism classes of non-isotrivial curves of genus \(g\) over \(L\) having good reduction in codimension 1. It is shown that \(C_g^2(L)\) is a finite set, and that there exists a number \(N_g^2(L)\) such that \(|X(L)|\leq N_g^2(L)\) for any curve \(X\in C_g^2(L)\) (theorem 3).
The proofs of the first two theorems are based on the results from an earlier paper [L. Caporaso, Compos. Math. 130, 1-19 (2002; Zbl 1067.14022)]. The proof of theorem 3 uses moduli maps. Note that if \( \dim V=1\) then theorem 3 implies a theorem of A. N. Parshin [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191-1219 (1968; Zbl 0181.23902)].
The author discusses an example suggested by J. de Jong, which shows that the cardinality of the set of fibrations with fixed degeneracy locus is not bounded as the cardinality of the degeneracy locus grows. The paper contains also some results on uniform boundedness of rational points which are independent of the degeneracy locus. It is concluded by introducing the notion of modular degree of a curve \(X\) and by proving some properties which relate this notion to \(|X(L)|\) and to the geometric Lang conjecture on the distribution of rational points on varieties of general type.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H05 Algebraic functions and function fields in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14G05 Rational points
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References:

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