A Schubert class in the Grassmannian is rigid if the only proper subvarieties representing that class are Schubert varieties. The hyperplane class σ1 is not rigid because a codimension one Schubert cycle can be deformed to a smooth hyperplane section. In this paper, we show that this phenomenon accounts for the failure of rigidity in Schubert classes. More precisely, we prove that a Schubert cl...

[1] The rapid unloading of ice from the southeastern sector of the Greenland ice sheet between 2001 and 2006 caused an elastic uplift of 35 mm at a GPS site in Kulusuk. Most of the uplift results from ice dynamic-induced volume losses on two nearby outlet glaciers. Volume loss from Helheim Glacier, calculated from sequential digital elevation models, contributes about 16 mm of the observed upli...

Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schuber...

A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F -regular if the section ring S(L) = ⊕n≥0 H (X,Ln) of an ample line bundle L on X is strongly F -regular in the sense of Hochster and Huneke [9] (cf. Definition 1.1). This important notion was introduced by Karen Smith in [18]. In this paper we prove that Schubert varieties are g...

We characterize by pattern avoidance the Schubert varieties for GLn which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varie...

CONTENTS We prove two lemmata about Schubert calculus on generalized 1. Background on Schubert Problems f|ag manifolds G/B, and in the case of the ordinary flag manifold 2. The Schubert Problems Graph and Its Structure for Small GLn/B we interpret them combinatorially in terms of descents, GLn(C) and geometrically in terms of missing subspaces. One of them 3. Proofs of the Lemmata gives a symme...

We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G{B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. These operators define a represe...

Formulating a Schubert problem as the solutions to a system of equations in either Plücker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation en...