Singular loci of varieties of complexes (Q1582743)

Let \(M(l\times m)\) be the affine space of matrices over a field with \(l\) rows and \(m\) columns. Denote by \(V\) the algebraic set consisting of those pairs \((A_1, A_2)\) in \(M(n_2\times n_1) \times M(n_3\times n_2)\) such that \(A_2A_1=0\). Then each irreducible component of \(V\) is isomorphic to the opposite cell in a Schubert variety \(SL(n_1+n_2+n_3)/Q\) where \(Q\) is a parabolic group. In this article the singular locus of each irreducible component of \(V\) is determined. A conjecture of Lakshmibai and Sandhya on how to write the singular locus of the associated Schubert variety \(X(\nu) =\overline{B\nu B} \pmod B\) of \(C\) as a union of varieties \(X(\lambda)\) is also proved.