On Schubert varieties in the flag manifold of Sl(n,\({\mathbb C})\)
In this paper we study the structure of the Schubert varieties in the flag manifold Fl(n) associated to Sl(n,\({\mathbb{C}})\). Two main results are obtained. First, a combinatorial algorithm is obtained whereby one can determine the singular locus of a Schubert variety. This is summarized in the following two theorems: Theorem I: Let \(X=X(s)\) be the Schubert variety in Fl(n) corresponding to the permutation s. Let \(A=A(s)\) be the set of all pairs (i,j) which satisfy (i) \(1\leq j<i\leq n\) and (ii) either \(\{s(1),...,s(j)\}\subset \{1,2,...,i-1\}\) or \(\{1,...,j\}\subset \{s(1),...,s(i-1)\}.\quad -\) Then X is \(non-\sin gular\quad \Leftrightarrow \quad \#A=co\dim X.\) Theorem II: Let \(X=X(s)\) be as above and let \(Y=Y(t)\) be contained in X. For every pair (k,m), \(1\leq k,m\leq n\) let \(B(k,m)=\emptyset\) if \(\#(\{t(1),...,t(k)\}\cap \{1,...,m\})\neq \#(\{s(1),...,s(k)\}\cap \{1,.\quad..,m\})\) while if \(\#(\{t(1),...,t(k)\}\cap \{1,...,m\})=\#(\{s(1),...,s(k)\}\cap \{1,..\quad.,m\})\) let \(B(k,m)=the\) set of all pairs (i,j) which satisfy (1) i\(>m\) and \(i\not\in (\{t(1),...,t(k)\}\cap \{m+1,...,n\})\) and (2) \(j\leq k\) and t(j)\(\leq m\). Let B be the union of all such B(k,m), \(1\leq k,m\leq n\). Then X is non- singular \(along Y\quad \Leftrightarrow \quad \#B=co\dim X.\) Secondly, a geometric description of the non-singular Schubert varieties is obtained. This result may be summarized as follows: A certain class of geometrically defined Schubert varieties may be constructed inductively by beginning with a Grassmannian and repeatedly forming a fibre bundle whose fibre is a Grassmannian and whose base space is a variety which has already been constructed during the induction. We call these repeated fibrations of Grassmannians. These Schubert varieties are all obviously non-singular from the construction. We then show that every non-singular Schubert variety in Fl(n) coincides with one of these, and consequently is a repeated fibration of Grassmannians.
scientific article; zbMATH DE number 3927107