Finite morphism between Fano hypersurfaces (Q2565955)

The author considers finite morphisms \(f : X \rightarrow Y\) between smooth Fano hypersurfaces in \(\mathbb{P}^{n+1}\) (\(n \geq 3\)) of degree \(d_X, d_Y\), respectively. He proves that either \(f\) is an isomorphism or \(d_Y \leq d_X/2\), or one has the exceptional case: \(n = 4, d_X = 5, d_Y = 3\). He also gives a precise bound of \(\deg(f)\) in terms of \(d_X\) and \(n\). This main result is in connection with Peternell's conjecture: for Fano manifolds \(X, Y\), suppose that \(b_2(X) = 1\) and there is a surjective morphism \(f : X \rightarrow Y\). Then Fano indices satisfies \(\text{index}(X) \leq \text{index}(Y)\). The main tool is Hurwitz type formula due to \textit{E. Amerik, M. Rovinsky} and \textit{A. Van de Ven} [Ann. Inst. Fourier 49, No.2, 405--415 (1999; Zbl 0923.14008)]. See also \textit{D. Sheppard}'s preprints [Morphisms from quintic threefolds to cubic threefolds are constant, \texttt{http://arxiv.org/math.AG/0302006}; Towards characterizing morphisms between high dimensional hypersurfaces, \texttt{http://arxiv.org/math.AG/0302005}] for other applications of the formula.