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On a problem of Noether-Lefschetz type. (English) Zbl 0929.14003

By a result of R. Lazarsfeld [in: Complete intersections, Lect. 1st Sess. CIME, Acireale 1983, Lect. Notes Math. 1092, 29-61 (1984; Zbl 0547.14009)] each morphism from the complex projective space \({\mathbb P}^n \) to a smooth projective variety \(Y\) is either constant or is a surjective finite morphism to \(Y = {\mathbb P}^n \).
Moreover, a result of K. H. Paranjape and V. Srinivas [Invent. Math. 98, 425-444 (1989; Zbl 0697.14037)] shows that a non-constant morphism from the smooth quadric \(X_2 \subset {\mathbb P}^{n+1} \) to a smooth complex variety \(Y\) must be surjective finite, and then either \(Y = {\mathbb P}^n \) or \(Y = X_2\). More generally, let \(f:X_d \rightarrow Y\) be a morphism from the smooth hypersurface \(X_d \subset {\mathbb P}^{n+1} \) to the smooth projective variety \(Y\). This paper shows that in cases \(n = 2,3\) the surjectiveness and finiteness of \(f\) imposes strong restrictions on \(f\), \(X_d\) and \(Y\): Let \(n = 2\), \(d \geq 4\), and let \(X_d\) be general. Then either \(Y = {\mathbb P}^2 \), or \(Y = X_d\) and \(f\) is an isomorphism (theorem 1.1). Let \(n = 3\), \(d \geq 2\), and let \(X = X_d\) be general, and assume that \(f: X_d \rightarrow Y\) is not an isomorphism. Then \(Y\) is either \({\mathbb P}^3\), or possibly \(Y\) may be also the Del Pezzo threefold \(V_5 = G(2,5) \cap {\mathbb P}^6\) or the Mukai-Umemura threefold \(V_{22}^s\) (theorem 1.2).
Reviewer: A.Iliev (Sofia)

MSC:

14E05 Rational and birational maps
14J45 Fano varieties
14J30 \(3\)-folds
14J70 Hypersurfaces and algebraic geometry
32H25 Picard-type theorems and generalizations for several complex variables
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