  In this article we use a theorem of Carlson and Griffiths and compute periods
of linear algebraic cycles inside the Fermat variety of even dimension $n$ and
degree $d$. As an application, for examples of $n$ and $d$ we prove that the
locus of hypersurfaces containing two linear cycles whose intersection is of
low dimension, is a reduced component of the Hodge locus in the underlying
parameter space. We also check the same statement for hypersurfaces containing
a complete intersection algebraic cycle. Our result confirms the Hodge
conjecture for Hodge cycles obtained by the monodromy of the homology class of
such algebraic cycles. This is known as the variational Hodge conjecture.
