Singularites generiques et quasi-resolutions des varietes de Schubert pour le groupe lineaire

Abstract

We determine explicitly the irreducible components of the singular locus of any Schubert variety for GL_n(K), K being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along these components. The case of covexillary Schubert varieties was solved in an earlier work of the author [Ann. Inst. Fourier vol. 51 fasc. 2 (2001), 375-393]. Here, we first exhibit some irreducible components of the singular locus of X_w, by describing the generic singularity along each of them. Let S_w be the union of these components. As mentioned above, the equality S_w = Sing X_w is known for covexillary varieties, and we base our proof of the general case on this result. More precisely, we study the exceptional locus of certain quasi-resolutions of a non-covexillary Schubert variety X_w, and we relate the intersection of these loci to S_w. Then, by induction on the dimension, we can establish the equality.