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

By A. Cortez. 2001

49 citations recorded by the source corpus.

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 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 Σ 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 Xw, and we relate the intersection of these loci to Σw. Then, by induction on the dimension, we can establish the equality.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc1e70726160d7b3d02b86b6c49d586ef3e8682f

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Forest-Like Permutations (2006, background): “Results of Cortez [10℄, and independently Manivel [17℄, show that 1324 and 213̄54 avoidan e is ne essary and su ient to hara terize whi h S hubert varieties are generi ally lo ally fa torial.”
GOVERNING SINGULARITIES OF SCHUBERT VARIETIES (2006, background): “Now let Φ be the embedding of v into w = 589716234 where the underlined positions indicate the embedding, which in symbols is given by φ1 = 1, φ2 = 4, φ3 = 5, φ4 = 6, φ5 = 8.”
Maximal singular loci of Schubert varieties in $SL(n)/B$ (2001, methodology): “These formulas have also been found by Manivel [26] and Cortez [14].”
Singularities of Generalized Richardson Varieties (2010, background): “W/WP index the Schubert varieties which form the irreducible components of the singular locus of Xu, see [3, 4, 7, 8, 23, 31, 34, 37].”