Degree of the generalized Plücker embedding of a Quot scheme and Quantum cohomology

Abstract

We compute the degree of the generalized Plücker embedding $\kappa$ of a Quot scheme $X$ over $\PP^1$. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from $\PP^1$ to the Grassmanian $\mathrm{Grass}(\mathrm{m},\mathrm{n})$. Then the degree of the embedded variety $\kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $\mathrm{Grass}(\mathrm{m},\mathrm{n})$ through the map $\psi$ that evaluates a map from $\PP^1$ at a point $x\in \PP^1$. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. 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 $\kappa (X)$.

18 pages, Latex document

References in corpus (2)

• Topological Mirrors and Quantum Rings
• Quantum and Floer cohomology have the same ring structure

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