Projection Cascades of models of log del Pezzo surfaces

TL;DR

This paper introduces the concept of type-I projection cascades for models of log del Pezzo surfaces, analyzing their existence and properties in weighted projective spaces, leading to new classifications of these algebraic surfaces.

Contribution

It defines and studies the existence of type-I projection cascades for log del Pezzo surfaces, providing explicit examples with cascades of lengths three and four.

Findings

Constructed cascades of length three and four.

Identified conditions for well-formed and quasismooth models.

Described images of models in weighted projective spaces.

Abstract

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.