##
**Higher Fano manifolds and rational surfaces.**
*(English)*
Zbl 1124.14039

If \(X\) is a Fano manifold, the pseudoindex of \(X\) is the minimal \(c_1(T_X)\) degree of a rational curve on \(X\). The Fano manifold \(X\) is called 2-Fano if the 2-nd graded piece of the Chern character \(\text{ch}_2(X) = 1/2(c_1(T_X)^2-2c_2(T_X))\) is nef. Examples of 2-Fano 3-folds are studied in more detail in the authors’ e-print [A note on Fano manifolds whose second Chern character is positive; arXiv:math.AG/0602644]. In particular, a smooth complete intersection \(X \subset {\mathbb P}^n\) of hypersurfaces of degrees \(d_1,\dots,d_s\) is Fano iff \(d_1+\cdots +d_s \leq n\), and \(X\) is 2-Fano iff \(d_1^2 + \cdots +d_s^2 \leq n\).

By Theorem 1.2, for a 2-Fano manifold \(X\) the irreducibility of some spaces of rational curves on \(X\) implies that the general point of \(X\) is contained in a rational surface. The proof uses the bend-and-break approach of Y. Miyaoka and S. Mori [Ann. Math. (2) 124, 65–69 (1986; Zbl 0606.14030)]. In Section 3 the authors prove that the conditions of Theorem 1.2 are fulfilled for certain Fano manifolds of pseudoindex 3. The main application of these results concerns the 2-Fano complete intersections in projective spaces. By Proposition 1.4, if \(X\) is any smooth 2-Fano complete intersection in \({\mathbb P}^n\) then the general point \(x \in X\) lies in a rational surface. Moreover, every irreducible component of the moduli space \(\bar{\mathcal M}_{o,o}(X)\) of genus 0 stable maps to \(X\), whose general point parametrizes a free curve [see C. Araujo and J. Kollár, Bolyai Soc. Math. Stud. 12, 13–68 (2003; Zbl 1080.14521); arXiv:math.AG/0203173], is uniruled.

In Section 4 are given 2 examples which show that the conditions of the results in the paper are sharp. The first example is the cubic fourfold \(X \subset {\mathbb P}^4\), which is Fano but not 2-Fano. By a recent result of the authors, for the general cubic 4-fold \(X\) any component of \(\bar{\mathcal M}_{o,o}(X)\) parametrizes a free curve, but infinitely many of these components are not uniruled [A. J. de Jong and J. Starr, Ill. J. Math. 48, No. 2, 415–450 (2004; Zbl 1081.14007)]. This shows that in Proposition 1.4 one can’t replace “2-Fano” by “Fano”.

By Theorem 1.2, for a 2-Fano manifold \(X\) the irreducibility of some spaces of rational curves on \(X\) implies that the general point of \(X\) is contained in a rational surface. The proof uses the bend-and-break approach of Y. Miyaoka and S. Mori [Ann. Math. (2) 124, 65–69 (1986; Zbl 0606.14030)]. In Section 3 the authors prove that the conditions of Theorem 1.2 are fulfilled for certain Fano manifolds of pseudoindex 3. The main application of these results concerns the 2-Fano complete intersections in projective spaces. By Proposition 1.4, if \(X\) is any smooth 2-Fano complete intersection in \({\mathbb P}^n\) then the general point \(x \in X\) lies in a rational surface. Moreover, every irreducible component of the moduli space \(\bar{\mathcal M}_{o,o}(X)\) of genus 0 stable maps to \(X\), whose general point parametrizes a free curve [see C. Araujo and J. Kollár, Bolyai Soc. Math. Stud. 12, 13–68 (2003; Zbl 1080.14521); arXiv:math.AG/0203173], is uniruled.

In Section 4 are given 2 examples which show that the conditions of the results in the paper are sharp. The first example is the cubic fourfold \(X \subset {\mathbb P}^4\), which is Fano but not 2-Fano. By a recent result of the authors, for the general cubic 4-fold \(X\) any component of \(\bar{\mathcal M}_{o,o}(X)\) parametrizes a free curve, but infinitely many of these components are not uniruled [A. J. de Jong and J. Starr, Ill. J. Math. 48, No. 2, 415–450 (2004; Zbl 1081.14007)]. This shows that in Proposition 1.4 one can’t replace “2-Fano” by “Fano”.

Reviewer: Atanas Iliev (Sofia)

### MSC:

14J45 | Fano varieties |

14H10 | Families, moduli of curves (algebraic) |

14J10 | Families, moduli, classification: algebraic theory |

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\textit{A. J. De Jong} and \textit{J. Starr}, Duke Math. J. 139, No. 1, 173--183 (2007; Zbl 1124.14039)

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