Governing singularities of Schubert varieties

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable” invariants of singularities, we geometrically prove that this governs (1) the -locus of a Schubert variety, and (2) which Schubert varieties are globally not . The prototypical case is “singular”; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45–52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan–Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205–220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 335 (2003) 3915–3945, MR 2004f:14071; A. Cortez, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003) 396–445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108, MR 2005f:14096; L. Manivel, Le lieu singulier des variétés de Schubert, Int. Math. Res. Not. 16 (2001) 849–871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan–Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.