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Towards Characterizing Morphisms between High Dimensional Hypersurfaces Main Results and Point of View

We prove the following theorem over an algebraically closed field of characteristic zero. Let f : X → Y be a nonconstant morphism of hypersurfaces in P, n ≥ 4. If Y is nonsingular and of general type, then there is a morphism F : P → P such that F |X = f and F (Y ) = X. As a corollary, we see that deg Y divides deg X with quotient m, and f is given by polynomials of degree m. Survey of the Literature Classically, algebraic geometry has sought to classify varieties. Recently focus has expanded to include the classification of morphisms between algebraic varieties. Our goal in this paper is to shed some light on what type of morphisms can occur between hypersurfaces. Let us give some results from the literature. In [11], Paranjape and Srinivas show that every nonconstant morphism between smooth quadric hypersurfaces in P is an isomorphism for n ≥ 4. Schuhmann shows in [12] that the degree of a morphism from a smooth hypersurface of degree d in P to a smooth quadric threefold is bounded from above in terms of d. For d = 3 she obtains a very good bound on the degree of possible morphisms and proves that every morphism from a smooth cubic threefold to a smooth quadric threefold is constant. Amerik proves in [1] that the degree of a morphism from a smooth hypersurface in P to a smooth quadric hypersurface in P is bounded from above for n ≥ 4 using a different argument than [12]. Beauville shows in [3] that every endomorphism of a smooth hypersurface in P of dimension at least 2 and degree at least 3 is an automorphism. For this he uses a Hurwitz-type inequality from [2]. We will generalize this inequality in Section 1. Main Results and Point of View The main result of this paper is the following Theorem, which we prove in Section 2. Theorem 2. Assume the base field is algebraically closed of characteristic zero. Let f : X → Y be a morphism of hypersurfaces in P, n ≥ 4. If Y is nonsingular and of general type, then there is a morphism F : P → P such that f = F |X and F(Y ) = X. The proof of Theorem 2, and our study of morphisms between hypersurfaces in general, relies on the following definition. 1