Resultants and Chow Forms via Exterior Syzygies

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k + 1 variables is the Chow form of P embedded by the e-th Veronese mapping in P with n = ( k+e k ) − 1. More generally, the Chow divisor of a k-cycle ∑ i ni[Vi] on projective space is defined to be ∑ i niDi, where Di is the Chow divisor of Vi. The Chow divisor of a sheaf F with k-dimensional support is the Chow divisor of the associated k-cycle of F . In this paper we will give a new expression for the Chow divisor and apply it to give explicit formulas in many new cases. Starting with a sheaf F on P, we use exterior algebra methods to define a canonical and effectively computable Chow complex of F on each Grassmannian of planes in P. If F has k-dimensional support, we show that the Chow form of F is the determinant of the Chow complex of F on the Grassmannian of planes of codimension k + 1. The Beilinson monad of F [Beilinson 1978] is the Chow complex of F on the Grassmannian of 0-planes (that is, on P itself.) In particular, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4, 6 or 8 in five variables. The easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables, the Chow form of the Veronese surface in P. Using the tangent bundle of P, conclude that it can be written in “Bézout form” (described below) as the Pfaffian of the matrix  0 [245] [345] [135] [045] [035] [145] [235]