AI 中文总结
研究有限域上恰当概型的奈龙 - 塞维里群，通过证明第一陈类映射，将\(\text{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}\)与特定扎里斯基局部平凡类群等同，得到有限域类似定理，无需半正规性和不可约性。
AI 中文摘要
设\(X\)是有限域\(k\)上的恰当既约概型，\(\ell\)是不同于\(\text{char}k\)的素数，记\(\overline{X}=X\times_{k}\overline{k}\)为其到\(k\)的代数闭包\(\overline{k}\)的基变换。若\(\text{H}^{2}_{\text{et}}(\overline{X},\mathbb{Z}_{\ell}(1))\)中的一个类在\(\overline{X}\)的一个扎里斯基开覆盖上消失，则称其为扎里斯基局部平凡的。我们证明第一陈类映射将\(\text{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}\)与在\(\text{H}^{2}_{\text{et}}(\overline{X},\mathbb{Q}_{\ell}(1))\)中像具有零权重的扎里斯基局部平凡类群等同起来。这是巴比耶里 - 维亚莱 - 罗森申 - 斯里尼瓦斯关于恰当半正规复簇定理的有限域类似物。在有限域情形下，既不需要半正规性也不需要不可约性。
英文摘要
Let $X$ be a proper reduced scheme over a finite field $k$, let $\ell$ be a prime different from $\operatorname{char} k$, and write $\overline{X}=X\times_{k}\overline{k}$ for its base change to an algebraic closure $\overline{k}$ of $k$. Call a class in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(1))$ Zariski-locally trivial if it vanishes on a Zariski-open cover of $\overline{X}$. We prove that the first Chern class map identifies $\operatorname{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}$ with the group of Zariski-locally trivial classes whose image in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Q}_{\ell}(1))$ has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale--Rosenschon--Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.
Comments27 pp, v2: Minor corrections, added Acknowledgment, Abstract renders correctly
