Stabilization of fields of meromorphic functions on neighborhoods of a rational curve

TL;DR

The paper proves a stabilization phenomenon for meromorphic function fields near a smooth rational curve $C\cong\mathbb{P}^1$ with positive self-intersection on a smooth complex surface $F$. It introduces the invariant $\tau(F,C)$, showing $\tau(F,C)\le 2$, and treats the two nontrivial cases $\tau(F,C)=1$ and $\tau(F,C)=2$ to establish that there exists a neighborhood $U\supset C$ such that every meromorphic function on any smaller connected neighborhood $V\supset C$ extends to $U$. The authors compile general ramification facts for meromorphic mappings, develop an algebro-geometric auxiliary result based on quadratic transformations, and employ discriminant-based arguments to handle the $\tau(F,C)=2$ case. Together, these yield a local rigidity result for meromorphic functions on neighborhoods of $C$, with implications for extension properties of meromorphic mappings on complex surfaces.

Abstract

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve $C$ with positive self-intersection. We prove that there exists a neighborhood $U\supset C$ such that any meromorphic function defined on a connected neighborhood of $C$ in $U$ can be extended to a meromorphic function on the entire $U$.

Figures (1)

• Figure 1: A sequence of three blow-ups of $\mathbb P\xspace^2$ and a sequence of quadratic transformations corresponding to them

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Proposition 3.4
• proof
• proof : End of the proof of Proposition \ref{['prop:trdeg1']}