Projective embedding of stably degenerating sequence of hyperbolic Riemann surfaces

TL;DR

This paper proves that the Kodaira embedding of a sequence of hyperbolic Riemann surfaces converges to the embedding of the limit space, including additional projective lines, as the surfaces degenerate.

Contribution

It establishes the projective embedding convergence for degenerating hyperbolic Riemann surfaces using Bergman space techniques.

Findings

Kodaira embeddings converge to the limit space embedding.

Extra complex projective lines appear in the limit.

Results apply to sequences of genus ≥ 2 curves.

Abstract

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