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Degree of the generalized Pl�cker embedding of a quot scheme and quantum cohomology

1998; Springer Nature; Volume: 311; Issue: 1 Linguagem: Inglês

10.1007/s002080050173

1432-1807

M.S. Ravi, Joachim Rosenthal, X. Wang,

Homotopy and Cohomology in Algebraic Topology

We compute the degree of the generalized Plücker embedding κ of a Quot scheme X over IP 1 .The space X can also be considered as a compactification of the space of algebraic maps of a fixed degree from IP 1 to the Grassmanian Grass(m, n).Then the degree of the embedded variety κ(X) can be interpreted as an intersection product of pullbacks of cohomology classes from Grass(m, n) through the map ψ that evaluates a map from IP 1 at a point x ∈ IP 1 .We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology [Vaf92] [Int91] [Wit93].We arrive at the degree by proving a version of the classical Pieri's formula on the variety X, using a cell decomposition of a space that lies in between X and κ(X).