On Pluri-canonical Systems of Arithmetic Surfaces

Let $S$ be a Dedekind scheme with perfect residue fields at closed points, let $f: X\rightarrow S$ be a minimal regular arithmetic surface of fibre genus at least $2$ and let $f': X'\rightarrow S$ be the canonical model of $f$. It is well known that $\omega_{X'/S}$ is relatively ample. In this paper we prove that $\omega_{X'/S}^{\otimes n}$ is relative very ample for all $n\geq 3$.