Count Geometric variety

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Summer minicourse Quang Dao - Vietnam, UMich - Flavors of Schubert Calculus

  • Enumerative Geometry: counting algebro-geometric objects satisfying some given conditions.
  • Motivating question: How many lines meet all 4 gievn lines in \(P^3\) space?
    • lines in \(P^3\) is 2-plane in 4 space hence Consider Gr(2,4).
  • Plucker embedding is an injective embedding of minors to a closed subvariety hence a projective variety.
    • Contrapositive(Matrix has rank at least k iff at least one minor has rank at least k)= None of the minors have rank at least k iff Matrix has rank less than k
    • Each minor is a determinant which gives an equation on coefficients of basis vectors.
    • Anna Brosowsky’s example: let k=2, n=3. Let X be the matrix of variables, with entries x_ij. Then a matrix with rank < 2 is a matrix with all 2x2 minors 0, i.e. satisfying the 3 minors being 0: x_11x_22 = x12x21, x12x23=x22x13, and x11x23 = x13x21
    • Affine Charts. Theorem on locally isomorphic. Also irreducible and smooth.
  • Schubert Cells
    • Fix a basis and complete flag. Every subpace in Gr(k,n) can be represented by a unique \(k \times n\) row echelon form.
    • Poisition is the column index of leading (from right to left) 1s. The following zeros form a Young diagram.
    • Partition (number of row entries)\(\lambda_i\) Corresponding to Position \((n-k+i-\lambda_i)\). Open in Schubert variety is Schubert cell (denote by \(\circ\)). Schubert cell closure in Zariski topology.
    • The boundary Schubert variety/ Schubert cell is a disjoint union of cells. This gives an affine stratification.
    • The cohomology class is independent of choice of complete flag as they are related by Gl(k) action.
    • Example: set of k-subspaces meeting a given space.

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