Projective embedding of stably degenerating sequences of hyperbolic Riemann surfaces

Abstract

Given a sequence of genus $g\ge 2$ curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature $-1$, we prove that the Kodaira embedding using an orthonormal basis of the Bergman space of sections of a pluricanonical bundle also converges to the embedding of the limit space together with extra complex projective lines.