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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.

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