Secant Varieties of the Varieties of Reducible Hypersurfaces in P n
M.V. Catalisano, A.V. Geramita, A. Gimigliano, B. Harbourne, J. Migliore, U. Nagel, Y.S. Shin
Abstract
Given the space
V=
P
(
d+n−1
n−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
-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.