Singular lines of trilinear forms

Abstract

We prove that an alternating e-form on a vector space over a quasi-algebraically closed field always has a singular ( e - 1 ) -dimensional subspace, provided that the dimension of the space is strictly greater than e. Here an ( e - 1 ) -dimensional subspace is called singular if pairing it with the e-form yields zero. By the theorem of Chevalley and Warning our result applies in particular to finite base fields. Our proof is most interesting in the case where e = 3 and the space has odd dimension n; then it involves a beautiful equivariant map from alternating trilinear forms to polynomials of degree n - 1 2 - 1 . We also give a sharp upper bound on the dimension of subspaces all of whose two-dimensional subspaces are singular for a non-degenerate trilinear form. In certain binomial dimensions the trilinear forms attaining this upper bound turn out to form a single orbit under the general linear group, and we classify their singular lines.