ar X iv : m at h / 05 06 22 3 v 2 [ m at h . A C ] 1 3 Se p 20 05 COMBINATORIAL SECANT VARIETIES

ar X iv : m at h / 06 11 69 6 v 2 [ m at h . A C ] 1 3 A pr 2 00 7 PROLONGATIONS AND COMPUTATIONAL ALGEBRA

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations which are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model i...

ar X iv : m at h / 05 05 06 3 v 2 [ m at h . SG ] 2 5 Se p 20 06 1 A symplectic approach to van den Ban ’ s convexity theorem

Let G be a complex semisimple Lie group and τ a complex antilinear involution that commutes with a Cartan involution. If H denotes the connected subgroup of τ-fixed points in G, and K is maximally compact, each H-orbit in G/K can be equipped with a Poisson structure as described by Evens and Lu. We consider sym-plectic leaves of certain such H-orbits with a natural Hamiltonian torus action. A s...

ar X iv : m at h / 06 05 37 3 v 1 [ m at h . A G ] 1 5 M ay 2 00 6 FLATNESS OF THE LINKED GRASSMANNIAN

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.

ar X iv : m at h / 06 05 35 5 v 1 [ m at h . G R ] 1 3 M ay 2 00 6 Axes in Outer Space

ar X iv : m at h / 05 02 36 3 v 1 [ m at h . C O ] 1 6 Fe b 20 05 CHAINS IN THE BRUHAT ORDER

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Litt...