Let $\cX$ be a curve of genus $g\ge2$ and zero $p$-rank over an algebraically closed field of odd characteristic $p$. We classify the pairs $(\cX,G)$ for which $G\le\Aut(\cX)$ has no common fixed point and $|G|>24g(g-1)$. For $g\ge4$, the curves are explicit cyclic covers of the projective line, the Hermitian curve, or the Ree curve. We determine their full automorphism groups and the possible subgroups $G$, including the central extensions. The exceptional $A_7$ action in characteristic $5$ determines the Hermitian curve of degree $6$. The cases of genus $2$ and $3$ are treated separately. These results give an odd-characteristic counterpart to the classification of large automorphism groups of zero $2$-rank curves. As an application, we obtain a new characterization of the generalized Suzuki curve in which the fixed-point hypothesis is no longer required.
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, o...
Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchm\'aros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-r...
Let $\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\ge2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\Aut(\mathcal{X})$ of order $N\ge2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two...
Let $q=p^h$ be an odd prime power and let $X/\F_{q^2}$ be a maximal curve of genus at least two. We classify the curves for which the full geometric automorphism group is transitive on the set of $\F_{q^2}$-rational points. We prove that, if $h>1$, then $X$ is the Hermitian curve. If $h=1$, the only additional possibil...
Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality...
Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\mathrm{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\math...
Diego Tuzzolo· 0 citations
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