  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.
