Klein, Lie, and their early Work on Quartic Surfaces

TL;DR

This paper explores the early work of Klein and Lie on quartic surfaces, highlighting their contributions to line complexes, ruled quartics, and properties of Kummer surfaces, based on archival and published sources from 1869-1872.

Contribution

It uncovers new insights into Klein and Lie's early research on quartic surfaces and their development of line-to-sphere transformations and properties of asymptotic curves.

Findings

New results on Steiner, Plücker, and Kummer quartic surfaces

Lie's line-to-sphere transformation revealed surprising properties

Enhanced understanding of ruled quartics and asymptotic curves

Abstract

Special types of quartic surfaces were much studied objects during the 1860s. Quartics were thus very much in the air when Sophus Lie and Felix Klein first met in Berlin in 1869. As this study shows, such surfaces played a major role in their subsequent work, much of which centered on linear and quadratic line complexes. This mutual interest led them to a number of new results on the quartic surfaces of Steiner, Pl\"ucker, and Kummer, as well as various types of ruled quartics studied earlier by Cremona. This paper, which draws on unpublished archival sources as well as published work from the period 1869-1872, underscores the importance of this aspect of the early geometrical work of these two famous figures. A highlight was Lie's line-to-sphere transformation, which led to surprising new findings on properties of asymptotic curves on Kummer surfaces.