Singular lines of trilinear forms

J. Draisma, R. Shaw

Research output: Book/ReportReportAcademic

Abstract

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-formyields 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.
Original languageEnglish
Publishers.n.
Number of pages9
Publication statusPublished - 2009

Publication series

NamearXiv.org [math.AG]
Volume0909.5676

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