Chow forms of congruences

Abstract

For X PN an n-dimensional variety the set of linear spaces of dimension N − n − 1 meeting X defines a hypersurface, H, in the Grassmann variety G(N − n,N + 1). The homogeneous form in the Pl¨ucker coordinates defining H or H itself is called the Chow form of X. This notion was defined by Cayley [A. Cayley, “On a new analytical representation of curves in space”, Q. J. Pure Appl. Math. 3, 225-236 (1860), and 5, 81-86 (1862); for a modern treatment see M. Green and I. Morrison, Duke Math. J. 53, 733-747 (1986; Zbl 0621.14028)]. In the present paper the authors study Chow forms of integral surfaces in G(2, 4) following the approach of M. Green and I. Morrison. Let V be a fixed 4-dimensional space and F P3 סP3, the flag variety parametrizing all chains V1 V3, where Vi is a subspace of V with dim Vi = i. F parametrizes the lines of G and to each integral surface Y in G there corresponds, in a natural way, an integral hypersurface X in F. The main result in this paper is a characterization of integral hypersurfaces X in F that are Chow forms of integral surfaces in G, in terms of some differential equations.