Oguiso's isomorphic quartic K3 surfaces are Cremona isomorphic via the classical cubo-cubic transformation, not by a projective automorphism.
On the automorphisms of hypersurfaces
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The cubo-cubic transformation and K3 surfaces
Oguiso's isomorphic quartic K3 surfaces are Cremona isomorphic via the classical cubo-cubic transformation, not by a projective automorphism.