  We give a new practical method for computing subvarieties of projective
hypersurfaces. By computing the periods of a given hypersurface X, we find
algebraic cohomology cycles on X. On well picked algebraic cycles, we can then
recover the equations of subvarieties of X that realize these cycles. In
practice, a bulk of the computations involve transcendental numbers and have to
be carried out with floating point numbers. However, if X is defined over
algebraic numbers then the coefficients of the equations of subvarieties can be
reconstructed as algebraic numbers. A symbolic computation then verifies the
results.
  As an illustration of the method, we compute generators of the Picard groups
of some quartic surfaces. A highlight of the method is that the Picard group
computations are proved to be correct despite the fact that the Picard numbers
of our examples are not extremal.
