Enumerative Geometry

-Review 01

• Stratification Affine- cohomology ring
• Schubert does not depend on the choice of flag.
• Grassmannian Gr(2,4)
  • Fix a falg \(p \subset l \subset H\) in \(P^3\).
• From Cohomology to point counting
  • What does the cup product mean?
  • Product translates to addition of points.
  • If the intersection is transversal, there is single count. If intersection is proper, there is multiplicity.
  • When are intersection of Schubert varieties transversal?
    • Kleiman’s Transversality theorem: Algebraic group acts transitively on a variety, a subvariety is generically transverse to another on an open dense set and is independent of the choice of flag. Proof Ref: David Eisenbud and Joe Harris. 3264 and all that A second course in algebraic geometry. Cambridge University Press, 2016.).

Schubert Class

• Ring Structure of \(H^*(Gr(k,n))\).
  • Intersection in complementary dimensions.
    • Complementary partitions, Opposite Flags
  • Pieri’s Rule
    • For a partition, Horizontal strip of length \(b\) attaches \(b\) more boxes such that:
      • no two boxes are in the same column
      • the resulting Young diagram is a partition.
    • Product Counts
      • Proof bounds the respective partitions based on intersection dimension counting.
      • Corollary: generates Schubert class.
      • Example: Giambelli’s formula.
• Geometric proof.
• Next time: how to multiply two classes? Littlewood Richardson Rule.

R

• William Fulton. Young tableaux: with applications to representation theory and geometry. 35. Cambridge University Press, 1997.
• Jake Levinson. “Schubert Calculus Mini-Course”. In: (2014).