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Exceptional hypersurfaces of endomorphisms of \(\mathbb{C} P^n\). (Hypersurfaces exceptionnelles des endomorphismes de \({\mathbb{C}}\text{P}(n)\).) (French) Zbl 0967.32022

A holomorphic map \(F:P^n(\mathbb{C})\to P^n(\mathbb{C})\) is called an endomorphism of the complex projective space if \(F\) is not a constant map. An endomorphism \(F\) is given by an ordered set \((f_0,\dots,f_n)\) of polynomials \(f_i (i=0,1, \dots,n)\) of \(n+1\) complex variables with same degree and \(\bigcap^n_{i=0}f_i^{-1} (0)=\{0\}\). Every endomorphism \(F\) is surjective and for every \(m\in P^n(\mathbb{C})\) the number of elements in \(F^{-1}(m)\) is finite. The degree of \(F\) is defined as the degree \(d\) of \(f\) and we have \(\#(F^{-1}(m))= d^n\) for generic point \(m\). A hypersurface \(H\subset P^n(\mathbb{C})\) is called exceptional for \(F\) if \(F^{-1} (H)=H\) holds.
Under the assumption that \(d\geq 2\), the author proves that if a hypersurface \(H\subset P^n(\mathbb{C})\) is exceptional for an endomorphism \(F\) with \(\deg H\geq 3\) then \(H\) is not a smooth hypersurface.

MSC:

32M99 Complex spaces with a group of automorphisms
14J70 Hypersurfaces and algebraic geometry
Full Text: DOI

References:

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