Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire

AbstractWe determine explicitly the irreducible components of the singular locus of any Schubert variety for GLn(K),K being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along each of them.The case of covexillary Schubert varieties was solved in an earlier work of the author [Ann. Inst. Fourier 51 (2) (2001) 375]. Here, we first exhibit some irreducible components of the singular locus of Xw, by describing the generic singularity along each of them. Let Σw be the union of these components. As mentioned above, the equality Σw=SingXw 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 Xw, and we relate the intersection of these loci to Σw. Then, by induction on the dimension, we can establish the equality