Under the assumption that the base field k has characteristic 0, we prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double β-polynomials of Fomin and Kirillov by computing the fundamental classes of schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.

Let $X$ be a minuscule Schubert variety. In this paper, we associate a quiver with $X$ and use the combinatorics of this quiver to describe all relative minimal models $\widehat{\pi}:{\widehat{X}}\to X$. We prove that all the morphisms $\widehat{\pi}$ are small and give a combinatorial criterion for $\widehat{X}$ to be smooth and thus a small resolution of $X$. We describe in this way all small resolutions of $X$. As another application of this description of relative minimal models, we obtain the following more intrinsic statement of the main result of Perrin, J. Algebra 294 (2005), 431–462. Let $\alpha\in A_1(X)$ be an effective 1-cycle class. Then the irreducible components of the scheme Hom$_{\alpha}(p^1,X)$ of morphisms from $\mathbb{P}^1$ to $X$ and of class $\alpha$ are indexed by the set: ${\mathfrak{ne}}(\alpha)=\{\beta\in A_1(\widehat{X}) \mid \beta$ is effective and $\widehat{\pi}_*\beta=\alpha\}$ which is independent of the choice of a relative minimal model $\widehat{X}$ of $X$.

Let ${\cal S}$(g,N,p) be the Siegel modular variety of principally polarized Abelian varieties of dimension g with a Γ0(p)-level structure and a full N-level structure (where p is a prime not dividing N[ges ]3 and Γ0(p) is the inverse image of a Borel subgroup of Sp(2g,${\Bbb F}$p) in Sp(2g,${\Bbb Z}$)). This variety has a natural integral model over ${\Bbb Z}$[1/N] which is not semi-stable over the prime p if g>1. Using the theory of local models of Rapoport–Zink, we construct a semi-stable integral model of ${\cal S}$(g,N,p) over ${\Bbb Z}$[1/N] for g=2 and g=3. For g=2, our construction differs from de Jong's one though the resulting model is the same.