发表机构
Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了曲线自同构群大 $p$-子群阶的尖锐边界及首个间隙，分类了等号情形下的三种新曲线，并证明了弱不等式足以推出边界分类，最后刻画了首个可允许层对应的普通曲线。
AI 中文摘要
设 $X$ 是特征为奇素数 $p$ 的代数闭域上亏格 $g\ge2$ 的曲线，$G=\mathrm{Aut}(X)$。Giulietti 和 Korchmaros 的一个定理表明，若 $G$ 不固定任何点，且 $p$-子群 $S\le G_P$ 满足 $|S|>\frac{p}{p-1}g$，则 $X$ 是经典的超椭圆曲线、Hermitian 曲线或 Ree 曲线之一。我们确定了尖锐边界及其下方的第一个间隙。在等号情形，除经典曲线外，恰好出现三种类型：普通曲线 $y^p-y=x+c/x$，零 $p$-秩曲线 $y^3=x^p-x$（其中 $p\equiv2\pmod3$），以及在特征 $3$ 下的普通亏格六族 $y^3-y=c\left(x+\frac1{x^3-x}\right)$。作为关键成分，我们分类了全自同构群不固定任何点且允许阶大于亏格的 $p$-子群的零 $p$-秩曲线。随后我们证明，当 $|S|>p$ 时，边界分类已可由较弱的不等式 $|S|>p(g-p+1)/(p-1)$ 推出。对于 $|S|\ge p^3$，$m=(p-1)|S|/p$ 上方的禁止区间宽度至少为 $p(p-1)/2$，而在 $m+(p^2-1)/2$ 下方至多还能出现一个亏格，且具有刚性分歧结构。最后，我们分类了第一个可允许层：它仅出现在 $|S|=p^2$ 时，所得的普通曲线可由二维 Artin--Schreier 空间完整描述。
英文摘要
Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves $y^p-y=x+c/x$, the zero-$p$-rank curves $y^3=x^p-x$ for $p\equiv2\pmod3$, and, in characteristic $3$, the ordinary genus-six family $y^3-y=c\left(x+\frac1{x^3-x}\right)$. As a key ingredient, we classify zero-$p$-rank curves whose full automorphism group fixes no point and which admit a $p$-subgroup of order larger than the genus. We then prove that, when $|S|>p$, the boundary classification already follows from the weaker inequality $|S|>p(g-p+1)/(p-1)$. For $|S|\ge p^3$, the forbidden interval above $m=(p-1)|S|/p$ has width at least $p(p-1)/2$, and below $m+(p^2-1)/2$ at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for $|S|=p^2$, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin--Schreier spaces.
