In Grassmannians, dimension 3 and codimension 3 classes are realizable by irreducible subvarieties exactly when b²≥ac, and in G(2,n) classes are realizable over Q exactly when their coefficients form a log-concave sequence with no internal zeros.
The Rigidity Problem in Orthogonal Grassmannians
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abstract
We classify rigid Schubert classes in orthogonal Grassmannians. More generally, given a representative $X$ of a Schubert class in an orthogonal Grassmannian, we give combinatorial conditions which guarantee that every linear space parametrized by $X$ meets a fixed linear space in the required dimension.
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Realization of Cohomology Classes in Grassmannians
In Grassmannians, dimension 3 and codimension 3 classes are realizable by irreducible subvarieties exactly when b²≥ac, and in G(2,n) classes are realizable over Q exactly when their coefficients form a log-concave sequence with no internal zeros.