Which Schubert varieties are local complete intersections? (Q2864126)

The main result of the paper is a characterization of those Schubert varieties for the general linear group that are local complete intersection (lci). The authors prove that a Schubert variety \(X_w\) associated to a permutation \(w \in S_m\) is lci if and only if \(w\) avoids the six patterns 53241, 52341, 52431, 35142, 42513, and 35162.NEWLINENEWLINEThe paper is organized as follows. Section 2 presents the basic definitions and set-up. Sections 3 and 4 prove that Schubert varieties associated to permutations avoiding the given patterns are lci. The proof involves showing that the variety \(X_w\) is lci at the identity point by identifying a minimal set generators for its defining ideal. This uses the notion of the \textit{essential set} of \(w\) introduced in [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)] and Schubert varieties \textit{defined by inclusions} from work of \textit{V. Gasharov} and \textit{V. Reiner} [J. Lond. Math. Soc., II. Ser. 66, No. 3, 550--562 (2002; Zbl 1064.14056)].NEWLINENEWLINESection 5 proves the necessity of the pattern avoidance. The strategy is to identify a collection of intervals \([u,v]\) in the Bruhat order for which the corresponding slice is not lci. The authors then show that if \(w\) contains one of the six given patterns, then \(w\) \textit{interval contains} one of the intervals \([u,v]\) above and they conclude that \(X_w\) is not lci.NEWLINENEWLINESection 6 contains a number of applications, including formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties. Section 7 concludes with a number of open questions.