Singular Lines of Trilinear Forms

We prove that an alternating e-form on a vector space over a quasialgebraically 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 2-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.