Let X be a complex hypersurface in a Pⁿ-bundle over a
curve C. Let C'→C be a Galois cover with group G. In this paper we
describe the C[G]-structure of $H^p,q$(X x$_{c}$ C C') provided that X x$_{c}$ C' is
either smooth or n = 3 and X x$_{c}$ C' has at most ADE singularities. As
an application we obtain a geometric proof for an upper bound by Páal
for the Mordell—Weil rank of an elliptic surface obtained by a Galois
base change of another elliptic surface.
