Singular loci of ladder determinantal varieties and Schubert varieties. (English) Zbl 1001.14019

Gonciulea, N.; Lakshmibai, V.

J. Algebra 229, No. 2, 463-497 (2000).

The authors give an interpretation of “ladder determinantal varieties” as “opposite cells” in a suitable Schubert variety. It is then possible to determine the singular loci of these varieties. This leads to a conjecture on the irreducible components of the singular locus of a Schubert variety in the flag variety.
The essentially novel results of this article appear in sections 7 and 8 where an explicit description of the singular locus of a ladder determinantal variety of mixed type is provided. The last section deals with a reformulation of a previously published conjecture about the irreducible components of the singular locus of a Schubert variety in the classical manifold of full flags.
The reference to this reviewer’s article, as it appears in the introduction, states: “An identification similar to that in theorem 1 for the case \(t_1=\cdots=t_l\) has also been obtained by Mulay”. This is somewhat misleading since the reviewer’s article [S. B. Mulay, Adv. Math. 74, No. 1, 1-30 (1989; Zbl 0693.14021)] contains an algorithm for identification (and more) can be deduced in a straightforward manner.

Reviewer: S.B.Mulay (Knoxville)

MSC:

14M12	Determinantal varieties
14M15	Grassmannians, Schubert varieties, flag manifolds
14B05	Singularities in algebraic geometry

Keywords:

ladder determinantal variety; full flag; Schubert variety; singular locus

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