AI 中文总结
本文研究费马三次曲面上的72个六线构型，证明其在自同构群下分为大小18与54的两个轨道，给出几何解释，并在数域与有限域上通过行列式特征标完成算术区分。
AI 中文摘要
六线构型（sixer）是光滑三次曲面上六条两两互不相交的直线构成的构型，等价于选取六条定义了吹下到$\u2119^2$的例外曲线。我们研究费马三次曲面$x_0^3+x_1^3+x_2^3+x_3^3=0$上的72个六线构型。\n 我们证明这些六线构型在费马三次曲面的自同构群作用下分裂为两个轨道，大小分别为18和54。我们通过对应的平面吹胀模型给出了这种分解的几何解释：两种轨道类型的代表元确定了$\u2119^2$中的六点构型，其射影自同构群的阶分别为36和12。这些群与对应六线构型的稳定子群等同，且与两个轨道的大小一致。\n 随后我们在$K=\u211a(ω)$（其中$ω^2+ω+1=0$）上计算了与两个轨道代表元相关的射影群，并通过行列式平方类特征标$δ_K:\u2162GL_2(K)⟶K^*/(K^*)^2$从算术上区分它们。对于大小为18和54的轨道，其像的$\u211d_2$维数分别为1和2。模13后，对应的有限像分别为$\u2162SL_2(\u211d_{13})$和$\u2162GL_2(\u211d_{13})$，且对于所有正规化三元组的选取，行列式特征标的区分性都保持成立。
英文摘要
A sixer is a configuration of six pairwise skew lines on a smooth cubic surface, equivalently a choice of six exceptional curves defining a blow-down to $\mathbb{P}^2$. We study the $72$ sixers on the Fermat cubic surface $x_0^3+x_1^3+x_2^3+x_3^3=0$. We show that these sixers split into two orbits under the automorphism group of the Fermat cubic, of sizes $18$ and $54$. We give a geometric interpretation of this decomposition through the corresponding plane blow-up models: representatives of the two orbit types determine six-point configurations in $\mathbb{P}^2$ whose projective automorphism groups have orders $36$ and $12$, respectively. These groups identify with the stabilizers of the corresponding sixers and recover the two orbit sizes. We then compute the projective groups associated with representatives of the two orbits over $K=\mathbb Q(ω)$, where $ω^2+ω+1=0$, and distinguish them arithmetically by the determinant square-class character $δ_K:\mathrm{PGL}_2(K)\longrightarrow K^*/(K^*)^2$. Its images have $\mathbb F_2$-dimensions $1$ and $2$ for the orbits of sizes $18$ and $54$, respectively. Modulo $13$, the corresponding finite images are $\mathrm{PSL}_2(\mathbb F_{13})$ and $\mathrm{PGL}_2(\mathbb F_{13})$, respectively, and the determinant-character distinction persists for all choices of normalization triple.
