Some classical formul{\ae} for curves and surfaces

TL;DR

This paper reviews classical formulas from the 19th century for counting planes with specific tangency and intersection properties with algebraic surfaces in projective space, highlighting historical computational methods.

Contribution

It presents a historical analysis of Salmon's classical computations of enumerative geometry problems related to surfaces and their tangent planes.

Findings

Enumerates the number of tritangent planes to a surface.

Counts planes intersecting a surface with a tacnode.

Highlights the significance of Salmon's methods in algebraic geometry.

Abstract

The goal of this text is to present the computation by Salmon, in the second half of the XIXth century, of various numbers enumerating planes with a prescribed tangency pattern with a sufficiently general surface $S$ in $\mathbf{P}^3$ (or, equivalently, of hyperplane sections of $S$ with prescribed singularities). Emblematic among these are the number of tritangent planes, and the number of planes cutting out a curve with a tacnode.