Resultants and Chow forms via exterior syzygies

Abstract:

Given a sheaf on a projective space ${\mathbf P}^n$, we define a sequence of canonical and effectively computable Chow complexes on the Grassmannians of planes in ${\mathbf P}^n$, generalizing the well-known Beilinson monad on ${\mathbf P}^n$. If the sheaf has dimension $k$, then the Chow form of the associated $k$-cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension $k+1$. Using the theory of vector bundles and the canonical nature of the complexes, 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 a polynomial formula for the resultant of five homogeneous forms of degree eight in five variables.