On Pluri-canonical Systems of Arithmetic Surfaces

Abstract

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

10 pages, no figures