We obtain a sharp lower bound for the volumes of purely 1-dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the Cartesian product of a compact Riemann surface with itself. This leads to a lower bound of the Seshadri number of the canonical line bundle of the Cartesian product with respect to the diagonal. As a consequence, we obtain an upper bound for the hyperbolic injectivity radii of compact Riemann surfaces of a fixed gonality. In particular, we obtain the limiting behavior of the gonalities of a tower of compact Riemann surfaces. We also give an application of our results to an invariant related to the ample cone of the symmetric product of a Riemann surface.
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