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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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References:
[1] Fujita, T., Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, (1990), Cambridge University Press Cambridge · Zbl 0743.14004
[2] Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2, (1984), Springer-Verlag Berlin · Zbl 0541.14005
[3] Fulton, W., Young tableaux. with applications to representation theory and geometry, London Mathematical Society Student Texts, vol. 35, (1997), Cambridge University Press Cambridge · Zbl 0878.14034
[4] Józefiak, T.; Lascoux, A.; Pragacz, P., Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Izv. Akad. Nauk SSSR, Ser. Mat., 45, 3, 662-673, (1981), (in Russian) · Zbl 0471.14028
[5] Laksov, D., Splitting algebras and Gysin homomorphisms, J. Commut. Algebra, 2, 3, 401-425, (2010) · Zbl 1237.14065
[6] Laksov, D.; Thorup, A., Schubert calculus on Grassmannians and exterior powers, Indiana Univ. Math. J., 58, 1, 283-300, (2009) · Zbl 1198.14052
[7] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, (1995), Oxford Science Publications, The Clarendon Press, Oxford University Press New York, with contributions by A. Zelevinsky · Zbl 0487.20007
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