Obrifold del Pezzo surfaces in ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ format

We construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in ${\mathbb{P}}^6(w_i)$. One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient ${\mathbb{P}}^6(w_i)$ are parameterized by positive integers. In the other type of models, weights of ${\mathbb{P}}^6(w_i)$ and Fano index, both are parameterized by the positive integers. The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$, which has codimension 4 in ${\mathbb{P}}^7$. We also give a formula for the Hilbert series of a generic weighted ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ variety, a key tool in these constructions.