## On the normal bundles of rational curves on Fano 3-folds.(English)Zbl 1253.14048

A smooth projective variety $$X$$ over $$\mathbb{C}$$ is rationally connected if any pair of points in $$X$$ lies in a rational curve [J. Kollár, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag. (1995; Zbl 0877.14012)]. Equivalently, $$X$$ is rationally connected if and only if there is a very free rational curve: $f: \mathbb{P}^1 \to X, f^*T_X \cong \bigoplus \mathcal{O}(a_i), a_i>0.$
A general deformation of a very free curve is a closed immersion $$i: C\cong \mathbb{P}^1 \to X$$. In this case the normal sheaf $$N_{C/X} \cong \bigoplus_{i=1}^{n-1} \mathcal{O}(a_i), a_1 \geq a_2 \geq \ldots \geq a_{n-1} \geq 1$$, where $$n$$ is the dimension of $$X$$. We say the curve $$C$$ is balanced if $$a_1-a_{n-1} \leq 1$$, otherwise unbalanced.
Let $$M\subset \overline{\mathcal{M}}_{0,0}(X, \beta)$$ be an irreducible component of the Kontsevich moduli space of genus $$0$$ curves on X. Assume that for a general member $$[C] \in M$$, the corresponding rational curve $$f: C \to X$$ is very free. We call $$M$$ a component of very free rational curves on X. Furthermore we call $$M$$ a balanced (resp. unbalanced) component if $$C$$ is balanced (resp. unbalanced). Note that by upper semi-continuity, the splitting type of the normal sheaf is the same for all general curves in $$M$$.
The paper under review proves the following.
Theorem 1.4. Let $$X$$ be a Fano threefold of Picard number one. If $$X$$ has an unbalanced component $$M$$ of very free rational curves, then $$X = \mathbb{P}^3$$ and $$M$$ is the space of conics on $$X$$.
The idea of the proof is the following. Assume a general very free curve $$C$$ has normal sheaf $$\mathcal{O}(a)\oplus \mathcal{O}(a+b), b>1$$. Then the deformation of the curve with $$a+1$$ points fixed sweeps out a surface $$\Sigma$$, which is essentially a Hirzebruch surface and $$C$$ is a section or $$\Sigma \cong \mathbb{P}^2$$ and $$C$$ is a conic. In the first case the author uses the classification of Fano $$3$$-folds of Picard number one [V. A. Iskovskikh and Yu. G. Prokhorov, Parshin, A. N. (ed.) et al., Algebraic geometry V: Fano varieties. Transl. from the Russian by Yu. G. Prokhorov and S. Tregub. Berlin: Springer. Encycl. Math. Sci. 47, 1–245 (1999; Zbl 0912.14013)] and studies the Abel-Jacobi map induced by the rulings of the surface to get a contradiction (when the intermediate Jacobian is trivial, which happens for two families of Fano $$3$$-folds, one need a different argument). In the latter case, one can prove the threefold is $$\mathbb{P}^3$$ and the curve is a conic.

### MSC:

 14M22 Rationally connected varieties 14D15 Formal methods and deformations in algebraic geometry

### Keywords:

rational curves; Fano threefolds

### Citations:

Zbl 0877.14012; Zbl 0912.14013
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