Let \({\mathcal {X}}\) be a (projective, algebraic, non-singular, absolutely irreducible) curve of genus g defined over an algebraically closed field K of characteristic \(p \ge 0\), and let q be a prime dividing the cardinality of \(\text{ Aut }({\mathcal {X}})\). We say that \({\mathcal {X}}\) is a q-curve. Homma proved that either \(q \le g+1\) or \(q = 2g+1\), and classified \((2g+1)\)-curves up to birational equivalence. In this note, we give the analogous classification for \((g+1)\)-curves, including a characterization of hyperelliptic \((g+1)\)-curves. Also, we provide the characterization of the full automorphism groups of q-curves for \(q= 2g+1, g+1\). Here, we make use of two different techniques: the former case is handled via a result by Vdovin bounding the size of abelian subgroups of finite simple groups, the second via the classification by Giulietti and Korchmáros of automorphism groups of curves of even genus. Finally, we give some partial results on the classification of q-curves for \(q = g, g-1\).

设\（{\ mathcal {X}} \）是在特征\（p \ ge 0 \）的代数闭合场K上定义的g族的（射影，代数，非奇异，绝对不可约）曲线，并令q是除\（\ text {Aut}（{\ mathcal {X}}）\）的基数的素数。我们说\（{\ mathcal {X}} \）是q曲线。Homma证明\（q \ le g + 1 \）或\（q = 2g + 1 \）并分类为\（（2g + 1）\） -可以达到双等式。在本注释中，我们给出\（（g + 1）\）的类似分类-曲线，包括超椭圆\（（g + 1）\）-曲线的特征。同样，我们提供了\（q = 2g + 1，g + 1 \）的q-曲线的全同构群的表征。在这里，我们使用两种不同的技术：前一种情况是通过Vdovin限制有限简单组的阿贝尔亚组大小的结果来处理的，第二种情况是通过Giulietti和Korchmáros对偶属曲线的自同构组的分类来进行处理的。最后，我们给出\（q = g，g-1 \​​）的q曲线分类的部分结果。