论文标题
部分可观测时空混沌系统的无模型预测
Plane curves with a large linear automorphism group in characteristic $p$
论文作者
论文摘要
令$ g $为三维投影组的子组$ \ mathrm {pgl}（3，q）$在有限字段$ \ mathbb {f} _q $ q $的有限字段定义，被视为$ \ mathrm {pglm {pgl}（3，k）$ k $的子组$ \ mathbb {f} _q $。对于七个非孢子，最大亚组$ g $ of $ \ mathrm {pgl}（3，q）$，我们调查了定义在$ g $不变的$ k $上定义的（投影，不可约）的平面曲线。对于每一个，我们计算$ g $ invariant曲线的最低度$ d（g）$，提供所有$ g $ - invariant曲线的分类，并确定所有$ g $ g $ invariant curves的频谱中的第一个差距$ \ varepsilon（g）$。我们表明，除非由$ g $唯一确定，否则学位$ d（g）$的曲线属于铅笔。我们还指出，$ g $ invariant $ d（g）$具有特定的几何特征，例如frobenius nonclassicality和$ \ mathbb {f} _ {q^i} $的异常变化。对于大多数平面曲线的示例，由$ \ mathrm {pgl}（3，q）$的大型子组留下，整个曲线的整个自动形态组是线性的，即$ \ mathrm {pgl}（pgl}（3，k）$的子组。尽管这似乎是一种普遍的行为，但我们表明，对于某些不可还原平面曲线也可能发生相反的情况，也就是说，曲线具有大量的线性自动形态，但其完整的自动形态组是非线性的。
Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$. For the seven nonsporadic, maximal subgroups $G$ of $\mathrm{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves defined over $K$ that are left invariant by $G$. For each, we compute the minimum degree $d(G)$ of $G$-invariant curves, provide a classification of all $G$-invariant curves of degree $d(G)$, and determine the first gap $\varepsilon(G)$ in the spectrum of the degrees of all $G$-invariant curves. We show that the curves of degree $d(G)$ belong to a pencil depending on $G$, unless they are uniquely determined by $G$. We also point out that $G$-invariant curves of degree $d(G)$ have particular geometric features such as Frobenius nonclassicality and an unusual variation of the number of $\mathbb{F}_{q^i}$-rational points. For most examples of plane curves left invariant by a large subgroup of $\mathrm{PGL}(3,q)$, the whole automorphism group of the curve is linear, i.e., a subgroup of $\mathrm{PGL}(3,K)$. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.
