Let X be a quartic hypersurface in P^4 with (no worse than) terminal singularities. The Grothendieck-Lefschetz theorem states that the Picard rank of X is 1, i.e. that every Cartier divisor on X is a hyperplane section of X. However, no such result holds for the group of Weil divisors of X when X is not factorial. Birational geometry can be used to understand some aspects of the topology of Fano 3-folds with mild (terminal Gorenstein) singularities. The MMP yields an "explicit" description of the lattice of Weil divisors and in particular a bound on its rank. If time permits, I will show that this "geometric motivation" of (non-)factoriality for Fano 3-folds carries some information on rationality/birational rigidity questions.