A terminal \({\mathbb Q}\)-factorial Fano variety \(V\) with Picard group \({\mathbb Z}\) is birationally rigid if \(V\) is birationally equivalent neither to another Fano variety of the same type, nor to a fibration with a smooth general fiber of Kodaira dimension \(-\infty\); and if in addition \(\text{Bir}(V) = \text{Aut}(V)\) then \(V\) is birationally superrigid, cf. def.1 and 2. Theorem 3 states that any cyclic triple covering \(X\) of \({\mathbb P}^{2n}\), \(n \geq 2\) branched over a hypersurface \(S\) of degree \(3n\) with at most ordinary double points (such \(X\) is a terminal \({\mathbb Q}\)-factorial Fano variety \(V\) with Picard group \({\mathbb Z}\)) is birationally superrigid and the group \(\text{Bir}(X)\) is finite. In particular any such \(X\) is not rational.
The above theorem describes in particular all smooth birationally superrigid cyclic triple covers of projective spaces, see remark 5. By theorem 15, a triple cover \(X \rightarrow {\mathbb P}^{2n}\), \(n \geq 2\) as in theorem 3 (i.e. with a ramification hypersurface \(S\) with at most ordinary double points) can’t be birationally equivalent to an elliptic fibration. But as seen in remark 17, if \(S\) has e.g. one ordinary triple point and \(n = 2\) then \(X\) is birationally equivalent to an elliptic fibration. Theorem 18, generalizing theorems 3 and 15, shows that if the ramification hypersurface \(S\) of the cyclic triple covering \(X \rightarrow {\mathbb P}^{2n}\) from theorem 3 admits besides ordinary double points also ordinary triple points then the conclusions of theorems 3 and 15 still hold, except in the case \(n = 2\) when \(S\) has a triple point and \(X\) is birational to an elliptic fibration by the construction given in remark 17.