Theorem of the existence of rational curve
In this post, we will discuss about a theorem which plays a crucial role in the proof of Hartshorne conjecture stated by Shigefumi Mori in [Mor79].
Theorem 1: (Collolary 7, [Mor70]):
Let $X$ be an n-dimensional non singular projective variety over an algebraic closed field $k$ such that tangent bundle $T_X$ is ample. Prove that:
i) $X$ contains a rational curve and any rational curve can be deformed as a cycle into a sum of rational curves $C$ such that $(C.K^{-1}_X)=n+1$.
ii) If $C$ ($\subset X$) is a rational curve such that $(C.K^{-1}_X)=n+1$, the resolution $f: \mathbb{P} \longrightarrow C$ is unramified and $f^{*}(T_{X|C}) = O(2) \oplus O(1)^{n-1}$.
Why is this theorem important ? In the proof of theorem 8, Mori construted a variety $Y$ which "parameterizing the slight modification $Z$ of maximal connected family of rational curves with minimal degree" and "rational curves with minimal degree" are those curves $C$ in our theorem. How did he do that ? First he fixed a closed point $P$ such that $f(P) =Q$ is smooth in $X$ and $i: P \longrightarrow Q$, the morphism $f$ attained by using theorem 1. Next, he denoted $V$ be the connected components of $Hom_k(\mathbb{P}^1,X;i)$ containing $[f]$. Then he used Chow's variety to parameterizing $V$ and $Y$ is the normalization of $\overline{\alpha (V)}$ where $\alpha$ is a morphism which turns a morphism $v \in V$ into cycle $v(\mathbb{P}^{1})$. After that, he successfully proved that $Y \cong \mathbb{P}(T^{*}_{Q,X}) \cong \mathbb{P}^{n-1}$ (8.1), and I will not go futher in how he constructed $Z$, but you should know that $Y$ has an important role in that construction. .
From those steps of Mori, we find that theorem 1 is essential for Theorem 8. Now, I will give a detailed proof of Theorem 1 (which was given by Mori in [Mor79]). First, there are some lemmas for this proof that we need to know:
Lemma 1: (Theorem 6 in [Mor79]):
Let $X$ be an n-dimensional non singular projective variety over an algebraic closed field $k$. If the anti-canonical divisor $K^{-1}_X$ is ample, then $X$ contains a rational curve.
Lemma 2: (V, ex. 2.6 in [Hart77]:
Every locally free sheaf of finite rank on $\mathbb{P}^{1}$ is isomorphic to a direct sum of invertible sheaves.
Lemma 3: (property of exterior power):
For vector bundles $E$ and $F$ over $X$. Then:
$\Lambda^{m}(E \oplus F) = \oplus_{m=k+l} \Lambda^{k}(E) \otimes \Lambda^{l}(F)$ and
Remark. Lemma 3 rises imediately from the equivalence property of exterior power in commutative algera. And from [Hart66] we have another statement: $E \oplus F$ is ample if and only if $E$ and $F$ is ample.
Proof of theorem 1
1) Lemma 1 implies the second part of i). For any rational curve $C$ in $X$, let $f: \mathbb{P}^{1} \longrightarrow C$ is a resolution (normalization). Then $f^{*}( T_X \otimes O_C)$ is ample because $T_X$ is ample and it is isomorphic to $O(a_1) \oplus O(a_2) \oplus ... \oplus O(a_n)$, hence by the statement from remark, $O(a_i)$ must be ample, hence $a_i > 0$. We see that $T_{\mathbb{P}^1} = O(2)$ (because $\omega (\mathbb{P}^{1}) = O(-2)$) and there is a non-zero morphism from $T_{\mathbb{P}^{1}}$ to $f^{*}( T_X \otimes O_C)$ which is induced by a natural map from $T_{\mathbb{P}^{1}}$ to $f^{*}(T_X)$. Hence, there exists $a_j \geq 2$. Hence, we can apply lemma 3: $K^{-1}_X |_{C} = \Lambda^{n} O(a_1) \otimes ... \otimes O(a_n) = O(a_1+...a_n)$, thus $(K^{-1}_X.C) =deg K^{-1}_X |_{C} \geq a_1+...+a_n \geq n+1$. And hence by theorem 4 in [Mor79], we have i) .
2) Assume that $(K^{-1}_X.C)=n+1$, hence $a_1=2$ and $a_i=1$ for all $i >1 $. Thus, the natural morphism from $T_{\mathbb{P}^{1}}$ to $f^{*}( T_X \otimes O_C)$ will be an injective and sends $T_{\mathbb{P}^{1}}$ to a direct summand of $f^{*}( T_X \otimes O_C)$. Hence, for all $P \in C$ closed, $T_{P,\mathbb{P}^1} \rightarrow T_{f(P), X}$ is injective, hence $B \rightarrow A$ is surjective with $B$ and $A$ are duals of $T_{P,\mathbb{P}^1}$ and $T_{f(P), X}$ respectly, which makes $\Omega_{B/A} = 0$. Thus, $f$ is unramified.
[Mor79] S.Mori, Projective Manifolds with Ample Tangent Bundles, Annals of mathematics, second series Vol. 110, No. 3 (Nov. 1979), pp. 593 - 606.
[Hart66] R.Hartshorne, Ample vecter bundle, Publications mathématiques de l’I.H.É.S., tome 29 (1966), p. 63-94.
[Hart77] R.Hartshorne, Algebraic geometry, Springer.
Theorem 1: (Collolary 7, [Mor70]):
Let $X$ be an n-dimensional non singular projective variety over an algebraic closed field $k$ such that tangent bundle $T_X$ is ample. Prove that:
i) $X$ contains a rational curve and any rational curve can be deformed as a cycle into a sum of rational curves $C$ such that $(C.K^{-1}_X)=n+1$.
ii) If $C$ ($\subset X$) is a rational curve such that $(C.K^{-1}_X)=n+1$, the resolution $f: \mathbb{P} \longrightarrow C$ is unramified and $f^{*}(T_{X|C}) = O(2) \oplus O(1)^{n-1}$.
Why is this theorem important ? In the proof of theorem 8, Mori construted a variety $Y$ which "parameterizing the slight modification $Z$ of maximal connected family of rational curves with minimal degree" and "rational curves with minimal degree" are those curves $C$ in our theorem. How did he do that ? First he fixed a closed point $P$ such that $f(P) =Q$ is smooth in $X$ and $i: P \longrightarrow Q$, the morphism $f$ attained by using theorem 1. Next, he denoted $V$ be the connected components of $Hom_k(\mathbb{P}^1,X;i)$ containing $[f]$. Then he used Chow's variety to parameterizing $V$ and $Y$ is the normalization of $\overline{\alpha (V)}$ where $\alpha$ is a morphism which turns a morphism $v \in V$ into cycle $v(\mathbb{P}^{1})$. After that, he successfully proved that $Y \cong \mathbb{P}(T^{*}_{Q,X}) \cong \mathbb{P}^{n-1}$ (8.1), and I will not go futher in how he constructed $Z$, but you should know that $Y$ has an important role in that construction. .
From those steps of Mori, we find that theorem 1 is essential for Theorem 8. Now, I will give a detailed proof of Theorem 1 (which was given by Mori in [Mor79]). First, there are some lemmas for this proof that we need to know:
Lemma 1: (Theorem 6 in [Mor79]):
Let $X$ be an n-dimensional non singular projective variety over an algebraic closed field $k$. If the anti-canonical divisor $K^{-1}_X$ is ample, then $X$ contains a rational curve.
Lemma 2: (V, ex. 2.6 in [Hart77]:
Every locally free sheaf of finite rank on $\mathbb{P}^{1}$ is isomorphic to a direct sum of invertible sheaves.
Lemma 3: (property of exterior power):
For vector bundles $E$ and $F$ over $X$. Then:
$\Lambda^{m}(E \oplus F) = \oplus_{m=k+l} \Lambda^{k}(E) \otimes \Lambda^{l}(F)$ and
Remark. Lemma 3 rises imediately from the equivalence property of exterior power in commutative algera. And from [Hart66] we have another statement: $E \oplus F$ is ample if and only if $E$ and $F$ is ample.
Proof of theorem 1
1) Lemma 1 implies the second part of i). For any rational curve $C$ in $X$, let $f: \mathbb{P}^{1} \longrightarrow C$ is a resolution (normalization). Then $f^{*}( T_X \otimes O_C)$ is ample because $T_X$ is ample and it is isomorphic to $O(a_1) \oplus O(a_2) \oplus ... \oplus O(a_n)$, hence by the statement from remark, $O(a_i)$ must be ample, hence $a_i > 0$. We see that $T_{\mathbb{P}^1} = O(2)$ (because $\omega (\mathbb{P}^{1}) = O(-2)$) and there is a non-zero morphism from $T_{\mathbb{P}^{1}}$ to $f^{*}( T_X \otimes O_C)$ which is induced by a natural map from $T_{\mathbb{P}^{1}}$ to $f^{*}(T_X)$. Hence, there exists $a_j \geq 2$. Hence, we can apply lemma 3: $K^{-1}_X |_{C} = \Lambda^{n} O(a_1) \otimes ... \otimes O(a_n) = O(a_1+...a_n)$, thus $(K^{-1}_X.C) =deg K^{-1}_X |_{C} \geq a_1+...+a_n \geq n+1$. And hence by theorem 4 in [Mor79], we have i) .
2) Assume that $(K^{-1}_X.C)=n+1$, hence $a_1=2$ and $a_i=1$ for all $i >1 $. Thus, the natural morphism from $T_{\mathbb{P}^{1}}$ to $f^{*}( T_X \otimes O_C)$ will be an injective and sends $T_{\mathbb{P}^{1}}$ to a direct summand of $f^{*}( T_X \otimes O_C)$. Hence, for all $P \in C$ closed, $T_{P,\mathbb{P}^1} \rightarrow T_{f(P), X}$ is injective, hence $B \rightarrow A$ is surjective with $B$ and $A$ are duals of $T_{P,\mathbb{P}^1}$ and $T_{f(P), X}$ respectly, which makes $\Omega_{B/A} = 0$. Thus, $f$ is unramified.
[Mor79] S.Mori, Projective Manifolds with Ample Tangent Bundles, Annals of mathematics, second series Vol. 110, No. 3 (Nov. 1979), pp. 593 - 606.
[Hart66] R.Hartshorne, Ample vecter bundle, Publications mathématiques de l’I.H.É.S., tome 29 (1966), p. 63-94.
[Hart77] R.Hartshorne, Algebraic geometry, Springer.
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