  We determine the cycle class of join algebraic cycles inside smooth
hypersurfaces by means of their periods. We show that being a join algebraic
cycle is equivalent to have its associated Artin Gorenstein algebra isomorphic
to the tensor product of the Artin Gorenstein algebras of each generating
cycle. This decomposition allows us to relate the quadratic fundamental form
associated to the Hodge loci of each algebraic cycle with the one associated to
their join. As a first application we study at a second order approximation
several Hodge loci of join algebraic cycles given as combinations of two linear
cycles (previously studied by Movasati, Dan, Kloosterman and the second
author), showing the non-smoothness in some new cases. Our main application is
to show the existence of fake linear cycles inside hypersurfaces of any
dimension and degree, given as join of $0$-dimensional fake linear cycles
inside hypersurfaces of $\mathbb{P}^1$ with all their closed points defined
over $\mathbb{Q}$. This shows that also for high degree hypersurfaces, there
are infinitely many (non smooth) Hodge loci whose Zariski tangent space has the
same codimension as the codimension of the component associated to a linear
cycle.
