Invariant hypersurfaces of endomorphisms of the projective 3-space

Abstract

We consider surjective endomorphisms f of degree > 1 on the projective n-space P n with n = 3, and f −1 -stable hypersurfaces V . We show that V is a hyperplane (i.e., deg(V ) = 1) but with four possible exceptions; it is conjectured that deg(V ) = 1 for any n ≥ 2; cf. [7], [3].Dedicated to Prof. Miyanishi on the occasion of his 70th birthday

Totally Invariant Divisors of non Trivial Endomorphisms of the Projective Space

“…and Höring proved that no irreducible hypersurface of P n of degree n is totally invariant (see [Hör17]). This last statement finishes the proof of the conjecture in P 3 thanks to Zhang who showed that no quadric of P 3 is totally invariant (see [Zha13,Thm 1.1.]). Our goal here is to prove the two following theorems.…”

Invariant hypersurfaces of endomorphisms of projective varieties

“…Remark 1.8 and Example 1.10). In Corollary 1.4, X is indeed a Fano manifold; but one would like to know more about the V and even expects X = P n and V be a hyperplane; see [30] and the references therein. Such an expectation is very hard to prove even in dimension three and proving the smoothness of V is the key, hence the relevance of Proposition 2.1 below.…”