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.
On the classification of singular cubic threefolds
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abstract
We classify combinations of isolated singularities that can occur on complex cubic threefolds generalizing analogous results for cubic surfaces due to Schl\"{a}fli and Bruce--Wall. In addition, we provide concise combinatorial description of the possible configurations of simple singularities: they essentially correspond to subgraphs of a certain graph.
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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.