Secant varieties of the varieties of reducible hypersurfaces in P ⁿ

Given the space V=P ^{(d+n−1n−1)−1} of forms of degree d in n variables, and given an integer ℓ>1 and a partition λ of d=d ₁ +⋯+d _r , 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 d ₁ ,…,d _r . 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.