Secant varieties of the varieties of reducible hypersurfaces in P^n

Given the space V of forms of degree d in n variables, and given an integer l >1and a partition \u3bb of d =d1+\ub7\ub7\ub7+dr, it is in general an open problem to obtain the dimensions of the (l 121)-secant varieties \u3c3_l(X_{n 121,\u3bb}) for the subvariety X_{n 121,\u3bb} 82V of hypersurfaces whose defining forms have a factorization into forms of degrees d1, ..., dr. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of \u3c3_l(X_{n 121,\u3bb}) )for any choice of parameters n, l and \u3bb. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., r=2), we also relate this problem to a conjecture by Fr\uf6berg on the Hilbert function of an ideal generated by general forms