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Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...

This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...