Given the space V of forms of degree d in n variables, and given an integer s and a partition of d=d1+....+dr$, it is in general an open problem to obtain the dimensions of the (s-1)-secant varieties W for the subvariety of 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 W for any choice of the partition and for any choice of the parameters n, s. 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, we also relate this problem to a conjecture by Froberg on the Hilbert function of an ideal generated by general forms.
|Titolo:||Secant varieties of the varieties of reducible hypersurfaces in P^n|
|Data di pubblicazione:||2019|
|Appare nelle tipologie:||01.01 - Articolo su rivista|