On the Geometry of Sixers on the Fermat Cubic Surface

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.