Every finite group action on a cubic threefold singular along a line, plane, or the chordal curve is linearizable; actions on cubics singular along a conic are generally not linearizable, though all such actions are unirational.
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Equivariant geometry of cubic threefolds with non-isolated singularities
Every finite group action on a cubic threefold singular along a line, plane, or the chordal curve is linearizable; actions on cubics singular along a conic are generally not linearizable, though all such actions are unirational.