A running example inspired by Klein-Mukai

• Heuristically, one computes $\Lambda \simeq \mathbf{Z}^{19}$ such that

  $$\Pi \Lambda \approx 0 \qquad \operatorname{Pic}(X^{al})|_B \subseteq \Lambda \overset{?}{\subseteq} \operatorname{Pic}X^{al}.$$

• We can compute $\operatorname{Aut}\Lambda$, the isomorphism class seems to be $F_{42} \times \operatorname{PGL}(2,7)$.
• No small rational curves: There are no lines, no conics, no twisted cubics.
• The "smallest" non-trivial curves that appear are smooth rational quartics.
• Lattice computations with $\Lambda$ predict that there are

  133056

  smooth rational quartics spanning $\Lambda$.