Biregular models of log Del Pezzo surfaces with rigid singularities

We construct biregular models of families of log Del Pezzo surfaces with rigid cyclic quotient singularities such that a general member in each family is wellformed and quasismooth. Each biregular model consists of infinite series of such families of surfaces; parameterized by the natural numbers $\mathbb{N}$. Each family in these models is represented by either a codimension 3 Pfaffian format modelled on the Pl\"ucker embedding of Gr(2,5) or a codimension 4 format modelled on the Segre embedding of \(\mathbb{P}^2 \times \mathbb{P}^2 \). In particular, we show the existence of two biregular models in codimension 4 which are bi parameterized, giving rise to an infinite series of models of families of log Del Pezzo surfaces. We identify those models of surfaces which do not admit a \(\mathbb {Q}\)-Gorenstein deformation to a toric variety.
Comment: 27 pages, Improved version following referee's comments. To appear in AMS's Math. of Comp