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Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of$d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensionsof the $\ell$-secant varieties $\sigma_\ell ({\mathbb X}_{n-1,\lambda})$ forthe subvariety ${\mathbb X}_{n-1,\lambda} \subset V$ of hypersurfaces whosedefining forms have a factorization into forms of degrees $d_1,\ldots,d_r$.Modifying a method from intersection theory, we relate this problem to thestudy of the Weak Lefschetz Property for a class of graded algebras, based onwhich we give a conjectural formula for the dimension of $\sigma_\ell({\mathbbX}_{n-1,\lambda})$ for any choice of parameters $n,\ell$ and $\lambda$. Thisconjecture gives a unifying framework subsuming all known results. Moreover, weunconditionally prove the formula in many cases, considerably extendingprevious results, as a consequence of which we verify many special cases ofpreviously posed conjectures for dimensions of secant varieties of Segrevarieties. In the special case of a partition with two parts (i.e., $r=2$), wealso relate this problem to a conjecture by Fr\"oberg on the Hilbert functionof an ideal generated by general forms.

Secant Varieties of the Varieties of Reducible Hypersurfaces in ${\mathbb P}^n$