Hypersurfaces with rational self-maps 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?
 A: 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 
