论文标题
在有限场上的曲线的最大曲线上
On the Maximum Gonality of a Curve over a Finite Field
论文作者
论文摘要
从曲线到投影线的非恒定$ k $ morphormism的最小程度，平稳的几何连接曲线是平稳的几何连接曲线的最小程度。通常，$ g \ ge 2 $的曲线的曲线最多为$ 2G-2 $。在有限场上，F.K.的结果1930年代的施密特可以用来证明gonality最多是$ g+1 $。通过几何和计算的混合，我们改善了这种界限：对于有限场上$ g \ ge 5 $的曲线，gonality最多是$ g $。对于属$ g = 3 $和$ g = 4 $，相同的结果与$ 217 $例外相同：有两条曲线属$ 4 $和gonality $ 5 $，$ 215 $ $ 3 $ $ 3 $和gonality $ 4 $。在其他论文中找到了$ 4 $的示例，我们在这里复制了它们的方程式；在补充材料中，我们提供了属的方程式 - $ 3 $示例。
The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can be used to prove that the gonality is at most $g+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \ge 5$ over a finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the same result holds with exactly $217$ exceptions: There are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples.
