A short proof establishes that G-equivariant smooth intersections of two quadrics in P^5_C are projectively G-linear if and only if they contain a G-invariant line.
Intermediate Jacobians and linearizability
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Equivariant intermediate Jacobians and intersections of two quadrics
A short proof establishes that G-equivariant smooth intersections of two quadrics in P^5_C are projectively G-linear if and only if they contain a G-invariant line.