In this paper, we investigate the problem of simultaneously generations of $r$-jets of $\omega_X\otimes L^{\otimes m}$ when $L$ is ample and base point free. It turns out that in this case, the bound of $m$ is optimistic, i.e. $m\geq f(r)=n+r+1$, which is linear in $r$. Our results work over arbitrary characteristics. We also treat the same problem when $X$ is singular where $\omega_X$ is replaced by the pushforward of canonical sheaves or the cohomology sheaves of dualizing complexes.

中文翻译：

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关于伴随丛射流的同时产生

在本文中，我们研究了当 $L$ 充足且无基点时同时生成 $\omega_X\otimes L^{\otimes m}$ 的 $r$-jets 的问题。事实证明，在这种情况下，$m$ 的边界是乐观的，即 $m\geq f(r)=n+r+1$，在 $r$ 中是线性的。我们的结果适用于任意特征。当 $X$ 是单数时，我们也处理同样的问题，其中 $\omega_X$ 被规范层的推进或对偶复合物的上同调层所取代。