Computing the Chow Variety of Quadratic Space Curves

The Chow variety, introduced in 1937 by Chow and van der Waerden [4], parameterizes algebraic cycles of any fixed dimension and degree in a projective space, each given by its Chow form. The case of curves in IP goes back to an 1848 paper by Cayley [3]. A fundamental problem, addressed by Green and Morrison [8] as well as Gel’fand, Kapranov and Zelevinsky [6, §4.3], is to describe the equations defining Chow varieties. We present a definitive computational solution for the smallest non-trivial case, namely for cycles of dimension 1 and degree 2 in IP. The Chow form of a cycle of degree 2 is a quadratic form in the Plücker coordinates of the Grassmannian G(2, 4) of lines in IP. Such a quadric in G(2, 4) represents the set of all lines that intersect the given cycle. Quadratic forms in Plücker coordinates form a projective space IP. The Chow variety we are interested in, denoted G(2, 2, 4), is the set of all Chow forms in that IP. The aim of this note is to make the concepts in [3,4,8] and [6, §4.3] completely explicit. We start with the 9-dimensional subvariety of IP whose points are the coisotropic quadrics in G(2, 4). By [6, §4.3, Theorem 3.14], this decomposes as the Chow variety and the variety of Hurwitz forms [9], representing lines that are tangent to a quadric surface in IP. Section 1 studies the ideal generated by the coisotropy conditions. We work in a polynomial ring in 20 variables, one for each quadratic Plücker monomial on G(2, 4) minus one for the Plücker relation. We derive the coisotropic ideal from the differential characterization of coisotropy. Proposition 1 exhibits the decomposition of this ideal into three minimal primes. In particular, this shows that the coisotropic ideal is radical, and it hence resolves