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 λ of d =d1+···+dr, it is in general an open problem to obtain the dimensions of the (l −1)-secant varieties σ_l(X_{n−1,λ}) for the subvariety X_{n−1,λ}⊂V 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 σ_l(X_{n−1,λ}) )for any choice of parameters n, l and λ. 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 Fröberg on the Hilbert function of an ideal generated by general forms.