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.

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关于某些 Calabi-Yau 和 hyperkähler 变体的投影正态性的评论

我们证明了一些具有平凡典型丛的品种的有效非常充足性和投影正态性的一些结果。在第一部分中，我们证明了具有平凡规范丛的规则光滑四折上的充足线丛的有效投影正态性结果。更准确地说，我们证明对于具有平凡规范丛的规则光滑四重，$A^{\otimes 15}$ 是 $A$ 充足的射影正态。在第二部分中，我们强调了在射影 hyperk\"ahler 变体的某些类别的已知示例（直至变形）上的大量和全局生成的线束的倍数的射影正态性。作为推论，我们表明除了维度 $4 中的两个极值情况$ 和 $6$，上面提到的例子中任何充足且全局生成的线性系统的一般曲线部分是非超椭圆的。