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Degree formula for Grassmann bundles. (English) Zbl 1327.14218
Summary: Let $$X$$ be a non-singular quasi-projective variety over a field, and let $$\mathcal{E}$$ be a vector bundle over $$X$$. Let $$\mathbb{G}_X(d, \mathcal{E})$$ be the Grassmann bundle of $$\mathcal{E}$$ over $$X$$ parametrizing corank $$d$$ subbundles of $$\mathcal{E}$$ with projection $$\pi : \mathbb{G}_X(d, \mathcal{E}) \to X$$, let $$\mathcal{Q} \leftarrow \pi^\ast \mathcal{E}$$ be the universal quotient bundle of rank $$d$$, and denote by $$\theta$$ the Plücker class of $$\mathbb{G}_X(d, \mathcal{E})$$, that is, the first Chern class of the Plücker line bundle, $$\det \mathcal{Q}$$. In this short note, a closed formula for the push-forward of powers of the Plücker class $$\theta$$ is given in terms of the Schur polynomials in Segre classes of $$\mathcal{E}$$, which yields a degree formula for $$\mathbb{G}_X(d, \mathcal{E})$$ with respect to $$\theta$$ when $$X$$ is projective and $$\wedge^d \mathcal{E}$$ is very ample.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 05E05 Symmetric functions and generalizations
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