Degree Formula for Grassmann Bundles

Abstract

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 $\mathcal E$, and denote by $\theta$ the Pl\"ucker class of $\mathbb G_X({d}, \mathcal E)$, that is, the first Chern class of the universal quotient bundle over $\mathbb G_X({d}, \mathcal E)$. In this short note, a closed formula for the push-forward of powers of $\theta$ 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.