Higher secant varieties of the minimal adjoint orbit

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

Download Full-text

Geometric aspects of self-adjoint Sturm–Liouville problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000105 ◽

2017 ◽

Vol 147 (6) ◽

pp. 1279-1295

Author(s):

Yicao Wang

Keyword(s):

Boundary Conditions ◽

Liouville Equation ◽

Adjoint Action ◽

Principal Type ◽

Unitary Matrices ◽

Adjoint Orbits ◽

Sturm Liouville ◽

Refined Classification ◽

Adjoint Orbit

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

Download Full-text

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽

10.3390/math6120314 ◽

2018 ◽

Vol 6 (12) ◽

pp. 314 ◽

Cited By ~ 8

Author(s):

Alessandra Bernardi ◽

Enrico Carlini ◽

Maria Catalisano ◽

Alessandro Gimigliano ◽

Alessandro Oneto

Keyword(s):

Algebraic Geometry ◽

Projective Variety ◽

Tensor Decomposition ◽

Tensor Rank ◽

Segre Variety ◽

Secant Varieties ◽

Open Problems ◽

Veronese Variety ◽

Strong Motivation

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

Download Full-text

Singularities and syzygies of secant varieties of nonsingular projective curves

Inventiones mathematicae ◽

10.1007/s00222-020-00976-5 ◽

2020 ◽

Vol 222 (2) ◽

pp. 615-665

Author(s):

Lawrence Ein ◽

Wenbo Niu ◽

Jinhyung Park

Keyword(s):

Secant Varieties ◽

Projective Curves

Download Full-text

ON THE EXACTNESS OF KOSTANT–KIRILLOV FORM AND THE SECOND COHOMOLOGY OF NILPOTENT ORBITS

International Journal of Mathematics ◽

10.1142/s0129167x12500863 ◽

2012 ◽

Vol 23 (08) ◽

pp. 1250086 ◽

Cited By ~ 1

Author(s):

INDRANIL BISWAS ◽

PRALAY CHATTERJEE

Keyword(s):

Lie Algebra ◽

Lie Group ◽

Nilpotent Orbits ◽

Semisimple Lie Group ◽

Simple Lie Algebra ◽

Adjoint Orbits ◽

Adjoint Orbit

We give a criterion for the Kostant–Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra.

Download Full-text