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

Abstract

Given the space $V={\mathbb{P}}^{\left(\genfrac{}{}{0ex}{}{d+n-1}{n-1}\right)-1}$ 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 dimensions of the $\ell$-secant varieties ${\sigma }_{\ell }({\mathbb{X}}_{n-1,\lambda })$ for the subvariety ${\mathbb{X}}_{n-1,\lambda }\subset V$ of hypersurfaces whose defining forms have a factorization into forms of degrees $d_1,\dots ,d_r$. 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 ${\sigma }_{\ell }({\mathbb{X}}_{n-1,\lambda })$ for any choice of parameters $n,\ell$ and $\lambda$. 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.

48 pages; corrected a typo in the statement of Proposition 7.2 and added short explanation. Appeared in J. Algebra

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