论文标题
用最小的$ a $ numbers实现Artin-Schreier封面的积极特征
Realizing Artin-Schreier Covers with Minimal $a$-numbers in Positive Characteristic
论文作者
论文摘要
假设$ x $是在特征性$ p> 0 $和$ b \ subset x $的代数封闭字段定义的平稳的投影连接曲线，这是有限的，可能是空的，一组点。 Booher和CAIS确定了$ \ Mathbf {Z}/p \ Mathbf {Z} $的$ a $ number的下限 - $ x $的封面带有branch locus $ b $。对于奇数$ p $，在大多数情况下，尚不清楚是否实现了此下限。在此注释中，当$ x $是普通的时，我们使用正式补丁将这个问题简化为有关$ \ Mathbf {z}/p \ Mathbf {z} $的计算问题 - 仿期线的封面。作为一个应用程序，当$ p = 3 $或$ p = 5 $时，对于任何普通曲线$ x $和任何选择$ b $时，我们证明，$ x $的Artin-Schreier封面已实现，带有分支机构$ b $。
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B \subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $\mathbf{Z}/p \mathbf{Z}$-cover of $X$ with branch locus $B$. For odd primes $p$, in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$-numbers of $\mathbf{Z}/p\mathbf{Z}$-covers of the affine line. As an application, when $p=3$ or $p=5$, for any ordinary curve $X$ and any choice of $B$, we prove that the lower bound is realized for Artin-Schreier covers of $X$ with branch locus $B$.
