Projection cascades of models of log del Pezzo surfaces

We introduce the notion of type-I projection cascade for a biregular model (infinite series) of log del Pezzo surfaces. We study the existence of type-I projection cascades for known classes of models of log del Pezzo surfaces, such that their images under their anti-canonical embeddings in some weighted projective space, can be described as codimension 4 and codimension 3 varieties. We obtain two cascades of length three and four cascades of length two, where each projection gives rise to a well-formed and quasismooth biregular model in lower codimension.