Biregular models of log Del Pezzo surfaces with rigid singularities

TL;DR

This paper constructs infinite series of biregular models for log Del Pezzo surfaces with rigid cyclic quotient singularities, using Pfaffian and Segre embeddings, and analyzes their deformation properties.

Contribution

It introduces new biregular models parameterized by natural numbers, expanding the classification of log Del Pezzo surfaces with specific singularities.

Findings

Existence of infinite series of biregular models in codimension 3 and 4.

Identification of models not deformable to toric varieties.

Construction of models using Pfaffian and Segre formats.

Abstract

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.