Cayley forms and self dual varieties

TL;DR

This paper characterizes Cayley forms as those for which the associated variety equals its dual, and explores the algebraic equations defining these forms, revealing quadratic and higher degree relations.

Contribution

It provides a new criterion for Cayley forms based on dual varieties and describes the algebraic equations defining the variety of Cayley forms.

Findings

F is a Cayley form iff Z equals its dual variety

The variety of generalized Cayley forms is defined by quadratic equations

Honest Cayley forms are not fully characterized by quadratic and cubic equations

Abstract

Generalized Chow forms were introduced by Cayley for the case of 3-space, their zero set on the Grassmannian G(1,3) is either the set Z of lines touching a given space curve (the case of a `honest' Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1,3). Our main result is that F is a Cayley form if and only if Z = G(1,3) \cap {F=0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F_0 + Q F_1 of F, with F_0, F_1 harmonic, such that the harmonic projection of the Cayley equation is identically zero. We also give new equations for honest Cayley forms, but show with some calculations that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.