On Pluri-canonical Systems of Arithmetic Surfaces

TL;DR

This paper proves that for minimal regular arithmetic surfaces with fiber genus at least 2, the n-th tensor power of the relative canonical sheaf is relatively very ample for all n≥3, extending known positivity properties.

Contribution

It establishes the relative very ampleness of higher tensor powers of the canonical sheaf on arithmetic surfaces, generalizing previous results about ampleness.

Findings

2e0; 2e0; 2e0; for all n 2e0; 3.

2e0; 2e0; 2e0; the canonical sheaf's tensor powers are relatively very ample.

2e0; 2e0; 2e0; for minimal regular arithmetic surfaces with fiber genus 2e0; 2.

Abstract

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 $n\geq 3$.