This is a visualization of the (real points of the) Fano surface of lines on the Fermat cubic threefold. The gray curve is the (real points of the) divisor of lines of the second type, i.e. lines whose normal bundle is \(\mathcal{O}(1)\oplus \mathcal{O}(-1)\). For most cubic threefolds, this divisor is smooth, but for the Fermat it has 30 irreducible components, each smooth of genus one. Of these, 10 have real points.

The surface was sampled in the Grassmannian \(\mathrm{Gr}(2,5)\), under its Plücker embedding, and projected down from a well-chosen chart to \(\mathbb{R}^3\) via PCA. Constructed with much help from Codex/GPT 5.4.