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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.

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

Citations:

Zbl 0744.00029
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