On Resolving Singularities of Plane Curves via a Theorem attributed to Clebsch

TL;DR

This paper explores the historical development of a key theorem in birational geometry related to resolving singularities of plane curves, highlighting contributions by Bertini, Klein, and others from the late 19th century.

Contribution

It clarifies the historical attribution and development of a central theorem in algebraic geometry, emphasizing Klein's role and the context of 19th-century mathematical discourse.

Findings

Reconstructed the circumstances of Klein's attribution to Clebsch.

Analyzed correspondence revealing Klein's influence on the theorem's attribution.

Provided historical insights into the development of birational geometry.

Abstract

This paper discusses a central theorem in birational geometry first proved by Eugenio Bertini in 1891. J.L. Coolidge described the main ideas behind Bertini's proof, but he attributed the theorem to Clebsch. He did so owing to a short note that Felix Klein appended to the republication of Bertini's article in 1894. The precise circumstances that led to Klein's intervention can be easily reconstructed from letters Klein exchanged with Max Noether, who was then completing work on the lengthy report he and Alexander Brill published on the history of algebraic functions [Brill/Noether 1894]. This correspondence sheds new light on Noether's deep concerns about the importance of this report in substantiating his own priority rights and larger intellectual legacy.