Standard models of degree $1$ del Pezzo fibrations

We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations ${ \pi \colon X \longrightarrow C }$ofdegree$1$andrelativePicardnumber$1$. 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.