# On the formal principle for curves on projective surfaces

## TL;DR

This paper demonstrates that the formal neighborhood of a rigid smooth curve with trivial normal bundle on a complex projective surface uniquely determines the surface's birational class.

## Contribution

It establishes a new criterion linking the formal completion along a curve to the birational classification of surfaces.

## Key findings

- Formal completion along the curve determines the surface's birational class.
- Rigidity and trivial normal bundle are key conditions.
- Provides a new tool for classifying complex projective surfaces.

## Abstract

We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.