Araujo, Carolina Rational curves of minimal degree and characterizations of projective spaces. (English) Zbl 1109.14032 Math. Ann. 335, No. 4, 937-951 (2006). Let \(X\) be a smooth complex projective variety; \(X\) is said to be uniruled if there exists a rational curve through every point of \(X\). If \(X\) is uniruled there exist irreducible components of the scheme RatCurves\(^n(X)\) such that the corresponding universal families dominate \(X\); such components are called dominating families of rational curves.A dominating family \(H\) is called minimal if for a general \(x \in X\) the subfamily \(H_x\) parametrizing curves of \(H\) through \(x\) is proper. If \(X\) is uniruled then on \(X\) there exists a minimal dominating family of rational curves (for instance one can take a dominating family of minimal degree with respect to some fixed ample line bundle).J.-M. Hwang and N. Mok recently [in: School on vanishing theorems and effective results in algebraic geometry. Lect. notes school Trieste, Italy, 2000. ICTP Lect. Notes 6, 335–393 (2001; Zbl 1086.14506)] started the study of the subvariety of tangent directions \({\mathcal C}_x \subset \mathbb P (T_xX)\), which is defined as the closure of the image of the tangent map \(\tau_x:H_x --> \mathbb P (T_xX)\), defined by sending a curve which is smooth at \(x\) to its tangent direction at \(x\); by results of S. Kebekus [J. Algebr. Geom. 11, 245–256 (2002; Zbl 1054.14035)] and J.-M. Hwang and N. Mok [Asian J. Math. 8, 51–64 (2004; Zbl 1072.14015)] the map \(\tau_x:H_x \to {\mathcal C}_x\) is the normalization.The paper under review studies varieties for which, for a general \(x \in X\), \({\mathcal C}_x\) is a union of linear \(d\)-dimensional subspaces of \(\mathbb P(T_xX)\), showing that in this case there exists a variety \(X'\), with a finite morphism onto \(X\), a dense open subset \(U^0\) of \(X'\) and a \(\mathbb P^{d+1}\)-bundle \(\varphi^0:U^0 \to T^0\) (for a more precise statement see Theorem 3.1 and the discussion above).In case \({\mathcal C}_x\) is a linear \(d\)-dimensional subspace of \(\mathbb P(T_xX)\) for a general \(x \in X\), then there exists a dense open subset \(U^0\) of \(X\) and a \(\mathbb P^{d+1}\)-bundle \(\varphi^0:U^0 \to T^0\) such that any curve of \(H\) meeting \(X^0\) is a line in a fiber of \(\varphi^0\) (Theorem 1.1).A first application of this result is a unified geometric proof of the characterization of the projective space as the only smooth complex projective variety \(X\) whose tangent bundle \(TX\) contains an ample locally free subsheaf \(E\).The case \(E \simeq TX\) is a celebrated result of S. Mori [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)]; the case in which \(E\) is a line bundle was considered by J. M. Wahl [Invent. Math. 72, 315–322 (1983; Zbl 0544.14013)] with completely different methods. F. Campana and T. Peternell [Manuscr. Math. 97, No. 1, 59–74 (1998; Zbl 0932.14024)] settled the cases in which rank \(E \geq \dim X-2\), and the proof was completed for every rank by M. Andreatta and J. A. Wiśniewski [Invent. Math. 146, 209–217 (2001; Zbl 1081.14060)]; this last proof relies on Mori’s and Wahl’s results.A second application of Theorem 1.1 is a characterization of products of projective spaces: Let \(X\) be a smooth complex projective variety with \(k\) distinct unsplit dominating families of rational curves \(H_1, \dots, H_k\) such that, for a general point \(x \in X\) the subvarieties of tangent directions at \(x\) to curves of the families are linear subspaces of dimension \(d_i\) with \(\sum d_i = \dim X -k\). Then \(X \simeq \mathbb P^{d_1 +1} \times \dots \times \mathbb P^{d_k +1}\). To prove this the author shows that, in the above assumptions, the numerical classes of the families \(H_1, \dots, H_k\) are linearly independent in \(N_1(X)\) and thus she can apply a result of the reviewer [Can. Math. Bull. 49, 270–280 (2006; Zbl 1115.14034)]. Reviewer: Gianluca Occhetta (Trento) Cited in 2 ReviewsCited in 20 Documents MSC: 14J40 \(n\)-folds (\(n>4\)) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli Citations:Zbl 1086.14506; Zbl 1054.14035; Zbl 1072.14015; Zbl 0423.14006; Zbl 0544.14013; Zbl 0932.14024; Zbl 1081.14060; Zbl 1115.14034 PDFBibTeX XMLCite \textit{C. Araujo}, Math. Ann. 335, No. 4, 937--951 (2006; Zbl 1109.14032) Full Text: DOI arXiv References: [1] Andreatta, M., Wiśniewski, J.A.: On manifolds whose tangent bundle contains an ample subbundle. Invent. 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