AbstractFor a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$
P
Σ
, the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$
i
∗
:
H
p
(
P
Σ
)
→
H
p
(
X
)
induced by the inclusion is injective for $$p=\dim X$$
p
=
dim
X
and an isomorphism for $$p<\dim X-1$$
p
<
dim
X
-
1
. This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$
NL
β
as the locus of quasi-smooth hypersurfaces of degree $$\beta $$
β
such that $$i^*$$
i
∗
acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$
dim
P
Σ
=
2
k
+
1
and $$k\beta -\beta _0=n\eta $$
k
β
-
β
0
=
n
η
, $$n\in \mathbb {N}$$
n
∈
N
, where $$\eta $$
η
is the class of a 0-regular ample divisor, and $$\beta _0$$
β
0
is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$
β
satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$
n
+
1
⩽
codim
Z
⩽
h
k
-
1
,
k
+
1
(
X
)
.
Deformations of smooth complete toric varieties: obstructions and the cup product
