Uniformly effective Shafarevich Conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus

The paper's main result is an effective uniform bound for the finiteness statement of the Shafarevich Conjecture over function fields. Several results on the projective geometry of curves are established in the course of the proof. These results should be of independent interest. As a corollary, a uniform effective bound for the Mordell Conjecture over function fields is derived via Parshin's trick.Comment: 19 page

On projectivized vector bundles and positive holomorphic sectional curvature

We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a K\"ahler metric with positive holomorphic sectional curvature