Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family.As sample applications, we show that the PGL 4 orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic threefold.
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Deopurkar et al. (2026) studied this question.
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