I'm looking for interesting examples of hypersurfaces $X\subset \mathbb P^n$ with a rational self-map $X\dashrightarrow X$?
Are there such examples for cubic hypersurfaces?
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4$\begingroup$ Every smooth cubic surface is the blow-up of a plane in 6 points, and therefore any birational transformation of the plane lifts to a rational self-map of the cubic surface $\endgroup$– Roberto PignatelliCommented Jul 30, 2014 at 11:57
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Most research in this area is about proving nonexistence of such rational maps for a sufficiently general hypersurface, other than the obvious rational maps: constant maps (which some disallow since they are not dominant) and the identity map. One of the people who studies this is Amerik. This is also related to birational superrigidity of general hypersurfaces, the evolution of the Iskovskikh-Manin method.
Of course every smooth cubic hypersurface has many such nontrivial birational automorphisms. For every point $p$ in $X$, for a general point $q$ in $X\setminus\{p\}$, the line $L$ spanned by $p$ and $q$ intersects $X$ in a third point $r$. There is a rational involution that sends $q$ to $r$.
Edit. I double-checked, and I believe the article of Amerik I was thinking of is actually the following.
MR1467127 (98h:14049) Reviewed
Amerik, Ekaterina(F-GREN-F)
Maps onto certain Fano threefolds. (English summary)
Doc. Math. 2 (1997), 195–211 (electronic).
14J45 (14E20)
There is a follow-up by Amerik-Rovinsky-van de Ven.
MR1697369 (2000f:14056) Reviewed
Amerik, E.(F-GREN-F); Rovinsky, M.(RS-IUM); Van de Ven, A.(NL-LEID)
A boundedness theorem for morphisms between threefolds. (English, French summary)
Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 405–415.
14J30
The final word seems to be the following article of Beauville.
MR1809497 (2002b:14053) Reviewed
Beauville, Arnaud(F-NICE-LD)
Endomorphisms of hypersurfaces and other manifolds.
Internat. Math. Res. Notices 2001, no. 1, 53–58.
14J70
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$\begingroup$ Thanks, this is very helpful. I'll take a look at the above papers. $\endgroup$ Commented Jul 30, 2014 at 15:24
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$\begingroup$ A related question for the cubic case: Are there also examples of birational automorphisms of infinite order? $\endgroup$ Commented Jul 30, 2014 at 15:57
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3$\begingroup$ @B.Wellington: "Are there also examples of birational automorphisms of infinite order?" Of course it depends on the field, whether there are many rational points, etc. However, in general, the composition of two of these birational involutions will be a birational automorphism of infinite order. $\endgroup$ Commented Jul 30, 2014 at 16:05