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
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 ,
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]
11/20 Raymond continues notes
11/27 Thanksgiving break
12/4 Raymond continues notes
Key objects: Grassmannians, flag varieties, partial flags. Schubert cells,
Schubert varieties, Plucker coordinates, incidence varieties.
Tautological bundles. Cohomology, relation to symmetric functions. Schubert polynomials.
Applications to enumerative problems. [B, BV, T, Knu]
Bott-Samelson varieties and resolution of singularities. [Brion]
Cluster structure and positivity. [FWZ, SSW]
Derived category of coherent sheaves. Exceptional collections. Braid group action. [Kap, Kuz, KT]
[B] S. Billey. Tutorial on Schubert varieties and Schubert calculus. slides
[BV] S. Billey, R. Vakil. Intersections of Schubert varieties and other permutation array schemes.
M. Brion. Lectures on the geometry of flag varieties.
[C] I. Coskun. Lectures on homogeneous varieties. lectures
[EH] D. Eisenbud, J. Harris. 3264 & All That.
Intersection Theory in Algebraic Geometry. book
[FWZ] S. Fomin, L. Williams, A. Zelevinsky. Introduction to Cluster Algebras.
[Kap] M. Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces.
Invent. Math. 92 (1988), no. 3, 479-508. file .
[KT] M. Khovanov, R. Thomas. Braid cobordisms, triangulated categories, and flag varieties.
[Knu] A. Knuntson. Schubert calculus and puzzles.
[Kuz]A. Kuznetsov. Semiorthogonal decompositions in algebraic geometry.
[SSW] K. Serhiyenko, M. Sherman-Bennett, L. Williams. Cluster structures in Schubert varieties in the Grassmannian.
[T] J. Tymoczko. Permutation actions on equivariant cohomology.