Standard models of degree $1$ del Pezzo fibrations

TL;DR

This paper constructs a canonical birational model for degree 1 del Pezzo fibrations with specific singularities and embeds it into a weighted projective space, extending the understanding of their structure in algebraic geometry.

Contribution

It introduces a standard birational model for degree 1 del Pezzo fibrations with Gorenstein canonical singularities and provides an embedding into a relative weighted projective space, including G-equivariant cases.

Findings

Constructed a standard birational model with Gorenstein canonical singularities.

Embedded the model into the relative weighted projective space (1,1,2,3).

Extended the construction to the G-equivariant category.

Abstract

We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations ${ \pi \colon X \longrightarrow C }$ of degree $1$ and relative Picard number $1$. We also embed the standard model into the relative weighted projective space $\mathbb{P}_C (1,1,2,3)$. Our construction works in the $G$-equivariant category where $G$ is a finite group.