论文标题
在有限场上平滑1周期
Smoothing of 1-cycles over finite fields
论文作者
论文摘要
令$ x $为在有限字段上定义的平滑投影品种。我们表明，$ x $上的任何代数$ 1 $ - 循环在合理上等同于$ 1 $ -CYCLE，这是$ \ Mathbb {z} $ - $ x $上的平滑曲线的线性组合。我们还证明了Poonen的Bertini Theorem在有限领域的广义版本。给定$ x $上的非常足够的线条捆绑$ \ MATHCAL {l} $和任意的线条捆绑$ \ Mathcal {M} $，此版本意味着存在$ \ Mathcal {m} \ otimes \ otimes \ Mathcal \ Mathcal {l}^{l}^{\ otimes d} $的全局部分，以供平稳
Let $X$ be a smooth projective variety defined over a finite field. We show that any algebraic $1$-cycle on $X$ is rationally equivalent to a smooth $1$-cycle, which is a $\mathbb{Z}$-linear combination of smooth curves on $X$. We also prove a generalized version of Poonen's Bertini theorem over finite fields. Given a very ample line bundle $\mathcal{L}$ on $X$ and an arbitrary line bundle $\mathcal{M}$, this version implies the existence of a global section of $\mathcal{M}\otimes \mathcal{L}^{\otimes d}$ for sufficiently large $d$ whose divisor is smooth.
