Ulrich Bundles on certain Grassmann Bundles Over Curves

Let $E$ be a slope-semistable vector bundle on smooth projective curve $C$. Assume, $\pi :G=G{r}_C(r,E)\to C$ denote the relative Grassmannian of rank $r$ quotients of $E$. We prove that for every very ample polarization $L=O_G(1)\otimes {\pi }^{\ast }A$ the polarized variety $(G,L)$ possess an Ulrich bundle. The construction globalizes homogeneous ulrich bundle on Grassmannian fibres. The remaining cohomology vanishing problem reduced to an acyclicity criterion on the base curve. We finally use Falting’s cohomological charaterization of semistability.