# Remarks on projective normality for certain Calabi-Yau and hyperk\"ahler varieties

## TL;DR

This paper establishes effective bounds for projective normality of ample line bundles on certain Calabi-Yau and hyperk"ahler varieties, advancing understanding of their embedding properties.

## Contribution

It proves explicit projective normality results for ample line bundles on regular smooth fourfolds with trivial canonical bundle and explores normality of line bundles on hyperk"ahler varieties.

## Key findings

- A^{ ensor 15} is projectively normal on regular smooth fourfolds with trivial canonical bundle.
- Most curve sections of ample, globally generated line bundles are non-hyperelliptic, except in two extremal cases.
- Provides effective bounds for projective normality in specific geometric contexts.

## Abstract

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of projective hyperk\"ahler varieties. As a corollary we show that excepting two extremal cases in dimensions $4$ and $6$, a general curve section of any ample and globally generated linear system on the above mentioned examples is non-hyperelliptic.