Periods of join algebraic cycles

Abstract

We show, for all $n\ge 2$ even and $d\ge 2+\frac{4}{n}$, that the moduli of smooth degree $d$ hypersurfaces of $\mathbb{P}^{n+1}$ contains infinitely many different Hodge loci whose Zariski tangent space has the same codimension as the Hodge locus of linear cycles. We construct the Hodge cycles determining those Hodge loci as joins of $0$-dimensional cycles inside hypersurfaces of $\mathbb{P}^1$ with all their closed points defined over $\mathbb{Q}$. In order to analyze the cycle classes of these algebraic cycles, we establish a general formula for the cycle class of the join of any two algebraic cycles inside smooth hypersurfaces, expressed in terms of their periods. Furthermore, we prove that an algebraic cycle is a join of algebraic cycles, if and only if, its associated Artin Gorenstein algebra is the tensor product of the Artin Gorenstein algebras associated to each generating cycle.