Informal algebraic geometry seminar

This is an informal seminar on algebraic geometry. In Fall 2020 we plan to look at various key concepts in algebraic geometry applied to flag varieties and their relatives.

The seminar meets at 3-4 on Fridays. Please email Erik Carlsson , Roger Casals or Eugene Gorsky for details.

Warm-up problem: How many lines intersect four given lines in general position in three-dimensional space?


10/2 Eugene: intro to projective spaces and Grassmannians notes

10/9 Roger: intro to cohomology and intersection theory notes

Exercise set 1

10/16 Erik: intro to Schubert calculus notes , problems , solutions

10/23 Ashleigh: homology of Grassmannians [C,EH]

10/30 Ashleigh continues notes

11/6 Yuze: Pieri and Giambelli rules [C,EH] notes

11/11 Eugene: recap on homology and cohomology notes

Exercise set 2

11/13 Raymond: equivariant cohomology and puzzles [Knu,T] notes

11/20 Raymond continues notes

11/27 Thanksgiving break

12/4 Raymond continues notes

Preliminary program

  1. Key objects: Grassmannians, flag varieties, partial flags. Schubert cells, Schubert varieties, Plucker coordinates, incidence varieties.
  2. Tautological bundles. Cohomology, relation to symmetric functions. Schubert polynomials. Applications to enumerative problems. [B, BV, T, Knu]
  3. Bott-Samelson varieties and resolution of singularities. [Brion]
  4. Cluster structure and positivity. [FWZ, SSW]
  5. Derived category of coherent sheaves. Exceptional collections. Braid group action. [Kap, Kuz, KT]


  1. [B] S. Billey. Tutorial on Schubert varieties and Schubert calculus. slides
  2. [BV] S. Billey, R. Vakil. Intersections of Schubert varieties and other permutation array schemes. arXiv:0502468
  3. [Brion] M. Brion. Lectures on the geometry of flag varieties. arXiv:math/0410240
  4. [C] I. Coskun. Lectures on homogeneous varieties. lectures
  5. [EH] D. Eisenbud, J. Harris. 3264 & All That. Intersection Theory in Algebraic Geometry. book
  6. [FWZ] S. Fomin, L. Williams, A. Zelevinsky. Introduction to Cluster Algebras. arXiv:1608.05735
  7. [Kap] M. Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479-508. file .
  8. [KT] M. Khovanov, R. Thomas. Braid cobordisms, triangulated categories, and flag varieties. arXiv:0609335 [Nicolle]
  9. [Knu] A. Knuntson. Schubert calculus and puzzles. notes
  10. [Kuz]A. Kuznetsov. Semiorthogonal decompositions in algebraic geometry. arXiv:1404.3143
  11. [SSW] K. Serhiyenko, M. Sherman-Bennett, L. Williams. Cluster structures in Schubert varieties in the Grassmannian. arXiv:1902.00807
  12. [T] J. Tymoczko. Permutation actions on equivariant cohomology. arXiv:0706.0460 [Raymond]