Double covers of smooth hyperquadrics as ample and very ample divisors

Let p:A \to Q be a double cover of a smooth quadric hypersurface Q^n \subset P^{n+1} branched alomg a smooth hypersurface. The paper deals with the problem of classisying pairs (X,L) where X is a smooth complex projective (n+1)-fold and L \in Pic(X) is either a very ample or an ample line bundle, whose complete linear system |L| contains a smooth element A as above. Complete answers are given for n \geq 3.