Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox

Abstract

The study of algebraic curves other than conics was revived by Julius Plücker in the 1830s. He succeeded in showing how the duality paradox could be resolved for such curves. The paradox is that a curve of degree n will seemingly have a dual of degree n(n−1) that will in its turn have a dual of degree n(n−1)(n(n−1)−1). But by duality the dual of the dual of a curve must be the original curve, which forces n(n−1)(n(n−1)−1)=n, an equation that is plainly false for n>2. Plücker observed that each double point on a curve lowers the degree of the dual by 2, and each cusp lowers the degree of the dual by 3. Moreover, the dual of a double point is a bitangent and of a cusp an inflection point. He showed that a non-singular curve of degree n has 3n(n−2) inflection points, so the degree of the double dual will be reduced by 9n(n−2). A simple calculation then shows that this degree would therefore be reduced to n, and the paradox resolved, if the original curve has $\frac{1}{2}n(n-2)(n^{2}-9)$ bitangents. Plücker could only conjecture this result, which was proved in 1850 by Jacobi, but he did investigate the special case of the 28 bitangents to a curve of degree 4, and showed that they could all be real.

Plücker then turned to experimental physics and the study of cathode rays, and the subject he had opened up was developed by Otto Hesse, who made more systematic use of homogeneous coordinates and used the eponymous Hessian to locate inflection points on curves. The work of Plücker and Hesse successfully established the subject of algebraic projective geometry.