# Complex planar curves homeomorphic to a line have at most four singular points

## TL;DR

This paper proves that complex planar curves homeomorphic to a line can have at most four singular points, with a unique degree five example if exactly four are present.

## Contribution

It establishes an upper bound on the number of singular points for such curves and characterizes the unique degree five case with four singularities.

## Key findings

- Curves homeomorphic to a line have at most four singular points.
- The degree five curve with four singular points is unique up to projective equivalence.
- If four singular points exist, the curve's degree is exactly five.

## Abstract

We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a projective equivalence.