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 Plücker 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.