Isomorphism of Multiprojective Bundles and Projective Towers

Abstract: We study when two projective bundles over two arbitrary smooth projective varieties of different dimensions can be isomorphic. We show that two multi-projective bundles (fibre product of projective bundles) over different projective spaces cannot be isomorphic, except in the trivial case. We also give necessary and sufficient conditions for the top varieties of two height 3 towers of projective bundles being isomorphic, under certain assumptions.