  In this article, we develop the geometry of canonical stratifications of the
spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the
Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define
and introduce a version of a gravity operad constructed for a class of moduli
spaces $\mathcal{M}_{0,n}$, equipped with a hidden holomorphic involution. This
additional symmetry is associated to a split quaternionic structure. We
introduce a categorical framework to present this object. Interaction between
the geometry, physics and the arithmetics are discussed. An important feature
is that 0-divisors of the split quaternion algebra imply additional singular
points, and lead to investigations concerning the geometry and mixed Hodge
structures.

Algebraic Geometry