Del Pezzo surfaces over dedekind schemes (following A. Corti's paper)

We prove a theorem that for any Del Pezzo surface $X_{K}$ over the fraction field $K$ of almost arbitrary discrete valuation ring $\mathcal O$, there is an integral model of this surface $X/\mathrm{Spec}(\mathcal O)$ such that it has terminal singularities of index $1$, the reduced and irreducible central fiber and very ample anticanonical divisor (for $d>2$) and very ample divisor $-2K_{X}$ (for $d=2$), where $d$ is a degree of $X_{K}$.