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.
Linearization problem for finite subgroups of the plane Cremona group
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abstract
We give a complete solution of the linearization problem in the plane Cremona group over an algebraically closed field of characteristic zero.
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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.