Singular lines of trilinear forms

TL;DR

This paper proves the existence of singular subspaces for alternating e-forms over certain fields, introduces a special equivariant map for e=3, and classifies singular lines in specific cases, advancing understanding of multilinear forms.

Contribution

It establishes the existence of singular subspaces for alternating e-forms over quasi-algebraically closed fields and classifies singular lines for special trilinear forms.

Findings

Existence of singular (e-1)-dimensional subspaces when dimension > e.

Construction of an equivariant map for e=3 and odd dimensions.

Classification of singular lines in certain binomial dimensions.

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-3)/2. 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.