On certain uniformity properties of curves over function fields (Q2781940)

Let \(B\) be a nonsingular projective complex curve. If \(S \subset B\) is a finite subset of points, then the Shafarevich conjecture proved by Parshin and Arakelov asserts that there exist only finitely many nonisotrivial families of smooth curves of genus \(g\geq 2\) over \(B \setminus S\). The Shafarevich conjecture implies the Mordell conjecture which says that a nonisotrivial curve of genus \(\geq 2\) over a function field \(K\) has only finitely many \(K\)-rational points. NEWLINENEWLINEThe author proves the uniform versions of Shafarevich and Mordell conjectures. More precisely, let \(F_g(B,S)\) be the set of equivalence classes of nonisotrivial fibrations \(f: X \longrightarrow B\) with \(X\) a smooth relatively minimal surface, and for all \(b\notin S\), \(X_b\) is a smooth curve of genus \(g\). Then for fixed \(g\geq 2, q\), and \(s\) there is a number \(P(g,q,s)\) such that \(| F_g(B,S)| \leq P(g,q,s)\) for any curve \(B\) of genus \(q\) and any subset \(S\subset B\) of cardinality \(s\). This is a uniform version of the Shafarevich conjecture. As a corollary, it implies the uniform version of Manin's theorem: NEWLINENEWLINELet \(g\geq 2\) and \(q\) be fixed, \(B\) be a curve of genus \(q\), \(S\subset B, | S| \leq s\), and let \(X\) be a nonisotrivial curve of genus \(g\) defined over \(K= \mathbb{C}(B)\) and having smooth fibers outside of \(S\). Then there exists an integer \(M(g,q,s)\) such that for the set of \(K\)-rational points \(X(K)\) on \(X\) we have \(| X(K)| \leq M(g,q,s)\). NEWLINENEWLINEFinally, the author obtains uniformity results generalizing the uniform versions of Shafarevich and Mordell conjectures to rational points on curves defined over function fields of any dimension. For the Mordell conjecture the result is as follows: NEWLINENEWLINELet \(g\geq 2\) and \(s\) be nonnegative integers, and \(h\in \mathbb{Q}[x]\) be a polynomial such that \(h(\mathbb{Z})\subset \mathbb{Z}\). Let \(M_{h(x)}\) be the moduli space of smooth projective varieties \(V\) having ample canonical line bundle \(K_V\), and such that \(\chi(K_V^{\otimes n})=h(n)\). Then there exists a number \(M(g,h,s)\) such that for any \(V\in M_{h(x)}\), for any closed \(T\subset V\) such that \(K_v^{\text{dim} V-1}\cdot T \leq s\) and for any nonisotrivial curve \(X\) of genus \(g\), defined over the function field \(K = K(V)\) and having smooth fibers away from \(T\), we have \(| X(K)| \leq M(g,h,s)\). NEWLINENEWLINEThe proofs of the results are based on combining a relative version of Parshin's proof of boundedness part in proving the Shafarevich conjecture with techniques from the modern theory of stable curves and their moduli.