A diagonal cubic surface $\sum_{i=0}^3a_iT_i^3=0$ is not minimal if, for example, $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$. This should be because there is an exceptional line $D$ such that no element in the Galois group takes $D$ to a distinct exceptional line $D'$ and $D\cap D'\neq \emptyset$. I can't get my head around how to find which lines are these, corresponding to say $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$?
2026-03-28 10:01:33.1774692093
Disjoint exceptional lines on non-minimal cubic surface
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