VARIETIES OF POTENTIALLY STRATIFIED KISIN AND BARSOTTI-TATE DEFORMATIONS

Let F be a unramified finite extension of Q(p) and (rho) over bar be an irreducible mod p two-dimensional representation of the absolute Galois group of F. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil-Kisin modules associated to certain families of potentially Barsotti-Tate deformations of (rho) over bar. We prove that this variety is a finite union of products of P-1. Moreover, it appears as an explicit closed connected subvariety of (P-1)[F:Q(p)]. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti-Tate deformations of (rho) over bar.