Injectivity radius and gonality of a compact Riemann surface (Q2879631)

The article presents a study of the relations between hyperbolic geometry and algebraic geometry of Riemann surfaces. One essential problem is the gonality stratification of the moduli space of a compact Riemann surface. The authors determine a sharp lower bound for the volumes of pure one-dimensional complex analytic subvarieties in some geodesic tubular neighborhood of the diagonal of the Cartesian product of a compact Riemann surface with itself. They obtain a lower bound of the Seshadri number of the canonical line bundle of the Cartesian product with respect to the diagonal and, moreover, an upper bound for the hyperbolic injectivity radii of compact Riemann surfaces of a fixed gonality. In particular they obtain the limiting behavior of the gonalities of a tower of compact Riemann surfaces. They present an application of these results to an invariant related to the ample cone of the symmetric product of a Riemann surface.