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**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.

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.

Reviewer: Zhiyu Tian (Pasadena)