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

Abstract

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.