Count Geometric variety
Please find linked:
- PBS-Youtube-Schubert Calculus:
- Sperner’s Lemma and Fixed point thm? Tensor contraction? Thomas Lam’s course Proof of matchings by contracting 0, 1?
- Fulton Young Tableaux
- Variations on a Theme of Schubert Calculus
- Aaron Pixton
- Thomas Lam
- Charlotte Chen (Grassmannian, Rep theory geometric side)
- Tasho Kaletha (Langlands, Lefschetz Principle)
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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- Swaraj Sridhar Pande refers to Speyer’s Algebraic Geometry course for interpreting the equations of Plucker embedding to be a projective variety.