For simply-laced Schubert varieties, factoriality, Q-factoriality, and a Betti number equality are equivalent, and the anticanonical line bundle is computed by an explicit coroot matrix.
The isomorphism problem for Schubert varieties
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abstract
Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis.
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An anticanonical perspective on G/P Schubert varieties
For simply-laced Schubert varieties, factoriality, Q-factoriality, and a Betti number equality are equivalent, and the anticanonical line bundle is computed by an explicit coroot matrix.