Illinois Journal of Mathematics

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

Jen-Chieh Hsiao

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 23 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1403534482

Digital Object Identifier
doi:10.1215/ijm/1403534482

Mathematical Reviews number (MathSciNet)
MR3224557

Zentralblatt MATH identifier
1312.14117

Subjects
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]

Citation

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. https://projecteuclid.org/euclid.ijm/1403534482


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References

  • W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge Univ. Press, Cambridge, 1993.
  • M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser, Boston, MA, 2005.
  • M. Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 33–85.
  • A. Conca and J. Herzog, Ladder determinantal rings have rational singularities, Adv. Math. 132 (1997), no. 1, 120–147.
  • A. Conca, Ladder determinantal rings, J. Pure Appl. Algebra 98 (1995), no. 2, 119–134.
  • A. Conca, Gorenstein ladder determinantal rings, J. Lond. Math. Soc. (2) 54 (1996), no. 3, 453–474.
  • W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420.
  • N. Gonciulea and V. Lakshmibai, Singular loci of ladder determinantal varieties and Schubert varieties, J. Algebra 229 (2000), no. 2, 463–497.
  • N. Gonciulea and C. Miller, Mixed ladder determinantal varieties, J. Algebra 231 (2000), no. 1, 104–137.
  • D. Glassbrenner and K. E. Smith, Singularities of certain ladder determinantal varieties, J. Pure Appl. Algebra 101 (1995), no. 1, 59–75.
  • N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981–996.
  • C. Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With an appendix by Melvin Hochster.
  • A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318.
  • N. Lauritzen, U. Raben-Pedersen and J. F. Thomsen, Global $F$-regularity of Schubert varieties with applications to $D$-modules, J. Amer. Math. Soc. 19 (2006), no. 2, 345–355 (electronic).
  • V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in ${\rm Sl}(n)/B$, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 1, 45–52.
  • E. Miller and B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer, New York, 2005.
  • A. Ramanathan, Schubert varieties are arithmetically Cohen–Macaulay, Invent. Math. 80 (1985), no. 2, 283–294.
  • K. E. Smith, $F$-rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159–180.
  • H. Úlfarsson and A. Woo, Which Scubert varieties are local complete intersections? Proc. Lond. Math. Soc. (3) 107 (2013), 1004–1052.
  • A. Woo and A. Yong, When is a Schubert variety Gorenstein? Adv. Math. 207 (2006), no. 1, 205–220.