Illinois Journal of Mathematics

On the $F$-rationality and cohomological properties of matrix Schubert varieties

Jen-Chieh Hsiao

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We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the $F$-rationality of matrix Schubert varieties. Although it is known that such varieties are $F$-regular (hence $F$-rational) by the global $F$-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott–Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.

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Illinois J. Math., Volume 57, Number 1 (2013), 1-15.

First available in Project Euclid: 23 June 2014

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Primary: 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14M10: Complete intersections [See also 13C40] 05E40: Combinatorial aspects of commutative algebra 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]


Hsiao, Jen-Chieh. On the $F$-rationality and cohomological properties of matrix Schubert varieties. Illinois J. Math. 57 (2013), no. 1, 1--15. doi:10.1215/ijm/1403534482.

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