We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon’s conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

We prove that, for q odd and n ≥ 3, the group G = On(q) · 2 is maximal in either the orthogonal group O2n(q) or the special orthogonal group SO2n(q). The group G corresponds to the stabilizer of a spread of lines of PG(2n − 1, q) in which some lines lie on a quadric, some are secant to the quadric and others are external to the quadric.

We provide a formula for variational quasi-Newton updates with multiple weighted secant equations. The derivation of the formula leads to a Sylvester equation in the correction matrix. Examples are given.

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality. AMS Subject Classification: 14N05.

An existence result on diagonal solutions of a linear matrix inequality is used to study diagonal Hurwitz and Schur stability and to derive the secant condition for systems with cyclic structure. © 2008 Elsevier B.V. All rights reserved.

Complex simple Lie algebras were classified by Cartan and Killing 100 years ago. Their proof proceeds by reducing the question to a combinatorical problem: the classification of irreducible root systems, and then performing the classification. We present a new proof of the classification via the projective geometry of homogeneous varieties. Our proof is constructive: we build a homogeneous spac...

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider iterates of the function m (x) = 1 I ...

If X is a smooth projective toric variety of dimension n we give explicit conditions on characters of the torus giving an embedding X →֒ Pr that guarantee dimSecX = 2n+ 1. We also give necessary and sufficient conditions for a general point of SecX to lie on a unique secant line when X is embedded into Pr using a complete linear system. For X of dimension 2 and 3 we give formulas for deg SecX in...