TY - JOUR

T1 - Degree formula for Grassmann bundles

AU - Kaji, Hajime

AU - Terasoma, Tomohide

N1 - Funding Information:
The first author is supported by JSPS KAKENHI Grant Number 25400053 . The second author is supported by JSPS KAKENHI Grant Number 15H02048 .

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.

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U2 - 10.1016/j.jpaa.2015.05.024

DO - 10.1016/j.jpaa.2015.05.024

M3 - Article

AN - SCOPUS:84940447256

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 -