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**Rational curves on Fano varieties.**
*(English)*
Zbl 0776.14012

Classification of irregular varieties, minimal models and abelian varieties, Proc. Conf., Trento/Italy 1990, Lect. Notes Math. 1515, 100-105 (1992).

[For the entire collection see Zbl 0744.00029.]

Let \(X\) be a smooth projective variety of dimension \(n>0\), over a field \(k\) with \(k=\bar k\) and \(\text{char} k=0\). The main result of the paper is that if \(X\) is a Fano variety (i.e. \(-K_ X\) is ample) and \(\text{Pic}(X)\cong\mathbb{Z}\), then for a generic pair of points \(x_ 1,x_ 2\in X\) there is a smooth rational curve \(C_{12}\) containing \(x_ 1,x_ 2\) and such that \(C_{12}\cdot(-K_ X)\leq n(n+1)\). – As a consequence, we have (corollary 1 in the paper) that \((-K_ X)^ n\leq n(n+1)^ n\), which in turns yields the following

Corollary 2: For every \(n>0\) there are only finitely many deformations types of smooth projective \(n\)-dimensional Fano varieties over \(k\) such that \(\text{Pic}(X)=\mathbb{Z}\).

The idea of the proof is to use previous results by Matsusaka and Kollár, which say that there are only finitely many deformation types of pairs \((X,H)\) where \(X\) is a smooth projective variety, \(H\) an ample divisor on \(X\) and the two highest coefficients of the Hilbert polynomial of \(X\) are bounded. In our case (by corollary 1) the two highest coefficients of the Hilbert polynomial of \(X\) are bounded since they are both \(=(-K_ X)^ n\) and so the conclusion follows.

Let \(X\) be a smooth projective variety of dimension \(n>0\), over a field \(k\) with \(k=\bar k\) and \(\text{char} k=0\). The main result of the paper is that if \(X\) is a Fano variety (i.e. \(-K_ X\) is ample) and \(\text{Pic}(X)\cong\mathbb{Z}\), then for a generic pair of points \(x_ 1,x_ 2\in X\) there is a smooth rational curve \(C_{12}\) containing \(x_ 1,x_ 2\) and such that \(C_{12}\cdot(-K_ X)\leq n(n+1)\). – As a consequence, we have (corollary 1 in the paper) that \((-K_ X)^ n\leq n(n+1)^ n\), which in turns yields the following

Corollary 2: For every \(n>0\) there are only finitely many deformations types of smooth projective \(n\)-dimensional Fano varieties over \(k\) such that \(\text{Pic}(X)=\mathbb{Z}\).

The idea of the proof is to use previous results by Matsusaka and Kollár, which say that there are only finitely many deformation types of pairs \((X,H)\) where \(X\) is a smooth projective variety, \(H\) an ample divisor on \(X\) and the two highest coefficients of the Hilbert polynomial of \(X\) are bounded. In our case (by corollary 1) the two highest coefficients of the Hilbert polynomial of \(X\) are bounded since they are both \(=(-K_ X)^ n\) and so the conclusion follows.

Reviewer: A.Gimigliano (Firenze)

### MSC:

14J45 | Fano varieties |

14M20 | Rational and unirational varieties |

14H45 | Special algebraic curves and curves of low genus |

13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |

14D15 | Formal methods and deformations in algebraic geometry |