### Abstract

Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let GX(d,E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π:GX(d,E)→X, let Q←π*E be the universal quotient bundle of rank d, and denote by θ the Plücker class of GX(d,E), that is, the first Chern class of the Plücker line bundle, det<>Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for GX(d,E) with respect to θ when X is projective and dE is very ample.

Original language | English |
---|---|

Pages (from-to) | 5426-5428 |

Number of pages | 3 |

Journal | Journal of Pure and Applied Algebra |

Volume | 219 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2015 Dec 1 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Pure and Applied Algebra*,

*219*(12), 5426-5428. https://doi.org/10.1016/j.jpaa.2015.05.024

**Degree formula for Grassmann bundles.** / Kaji, Hajime; Terasoma, Tomohide.

Research output: Contribution to journal › Article

*Journal of Pure and Applied Algebra*, vol. 219, no. 12, pp. 5426-5428. https://doi.org/10.1016/j.jpaa.2015.05.024

}

TY - JOUR

T1 - Degree formula for Grassmann bundles

AU - Kaji, Hajime

AU - Terasoma, Tomohide

PY - 2015/12/1

Y1 - 2015/12/1

N2 - Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let GX(d,E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π:GX(d,E)→X, let Q←π*E be the universal quotient bundle of rank d, and denote by θ the Plücker class of GX(d,E), that is, the first Chern class of the Plücker line bundle, det<>Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for GX(d,E) with respect to θ when X is projective and dE is very ample.

AB - Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let GX(d,E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π:GX(d,E)→X, let Q←π*E be the universal quotient bundle of rank d, and denote by θ the Plücker class of GX(d,E), that is, the first Chern class of the Plücker line bundle, det<>Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for GX(d,E) with respect to θ when X is projective and dE is very ample.

UR - http://www.scopus.com/inward/record.url?scp=84940447256&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940447256&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2015.05.024

DO - 10.1016/j.jpaa.2015.05.024

M3 - Article

VL - 219

SP - 5426

EP - 5428

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 12

ER -