Constructing projective bundle morphism
In [Arau06], Araujo proved an interesting theorem which after that, used for proving a generalization of Hartshrone's conjecture. That theorem is:
Theorem: Suppose $C_x$ is a $d$- dimensional linear subspace of $\mathbb{P}(T_{x}X)$ for a general point $x \in X$. There is a dense open subset $X^0$ of $X$ and a $\mathbb{P}^{d+1}$-bundle $\varphi: X^0 \rightarrow T^0$ such that any curve from a minimal covering of rational curves, denote $H$, meeting $X^0$ is a line on a fibre of $\varphi$.
For the application of this theorem, I will write later. Now, I will give a part of the construction of
$X^0$ and $\varphi$, for the second statement, you can see theorem 3.1 in [Arau06].
First, there is some notion and properties that reader should know. I will copy the open part of [Arau06] right here.
Let $X$ be a smooth complex projective variety and, assume that X is uniruled. Let $H$ be an irreducible component of $RatCurves^{n}(X)$. We say that $H$ is a covering family if the corresponding universal family dominates $X$. A covering family $H$ of rational curves on $X$ is called minimal if, for a general point $x \in X$, the subfamily of $H$ parametrizing curves through $x$ is proper.
Let $x \in X$ be a general point and denote by $H_x$ the normalization of the subscheme of $H$ parametrizing rational curves passing through $x$. Let $\pi_x: U_x \rightarrow H_x$ and $\eta_x: U_x \rightarrow X$ be the universal family morphisms, so that $U_x$ is normal and $\pi_x$ is a $\mathbb{P}^1$-bundle ([Kol96, II.2.12]). Denote by $locus(H_x)$ the closure of the image of $\eta_x$ (with the reduced scheme structure). We remark that a rational curve smooth at $x$ is parametrized by at most one element of $H_x$.
For the concepts of $RatCurves^{n}(X)$ and $Univ^{rc}(x,X)$ ( $U_x$ here is $Univ^{rc}(x,X)$, readers can c.f. [Kol96, II.2].
Now, we will come to two key lemmas for constructing $X^0$ and $T^0$.
Remark: There is a result in [Kol96, II.2.15, II.3.5] state that $X$ is smooth over $locus(H_x)$. And we can also obtain $D$ as $D_x = \overline{C}_x \subset T_{x}X$ with $\overline{C}_x$ correspond to $C_x$ in $\mathbb{P}(T_{x}X)$.
Lemma 1: Suppose that $C_x$ is a union of linear subspaces of $\mathbb{P}(T_{x}X)$ for
general $x \in X$, with $X$ is uniruled projective variety and $D$ be as defined above. Then $D$ is tangent to $locus(H_x)$ for a general point $x \in X$. In particular, $locus(H_x)$ is smooth at $x$.
Lemma 2: Suppose that $C_x$ is a union of $d$-dimensional linear subspaces of $\mathbb{P}(T_{x}X)$ for general $x \in X$, with $X% is uniruled projective variety. Then, for a general point $x \in X$, the normalization of $locus(H_x)$ is isomorphic to $\mathbb{P}^{d+1}$. Under this isomorphism, the rational curves on $locus(H_x)$ parametrized by $H_x$ come from lines on $\mathbb{P}^{d+1}$ passing through a fixed point.
Now the first lemma helps us to provide a map from $locus(H_x)$ to the tangent point $x$, and from lemma 2, we can image that in $X^0$ we need $locus(H_x)$ must be its germ at $x$. From the "general" property, there must be a open dense subset $U$ s.t the condition that it contain all $locus(H_x)$ as general fibres and then provide a morphism to the point $x$ in the set of "tangent point" $T$. We see that $X^0$ and $T^0$ will be $U$ and $T$. And hence, we can come to a morphism $\varphi: X^0 \rightarrow T^0$ (in [Arau06], Araujo argued that $\varphi$ obtained by using "Frobenius' theorem", but I don't know what it is).
This constructing is useful for futher studies in the generalization of Hartshorne's conjecture and there is a lovely proposition that can play an important role in a proof of a even more general version of Hartshorne's conjecture.
Propostion: Let $C_x, X, X_0, T_0$ and $\varphi$ as above. Then $d = n-1$ and $X \cong \mathbb{P}^n$ if and only if $dim T_0 = 0$.
Proof of proposition. If $dim T_0 = 0$, then $T_0$ is a point. $\varphi^{-1}(T^0) = X^0 \cong \mathbb{P}^{d+1}$, but $X^0$ is dense in a projective variety $X$ with dimension $n$, then $X^0$ will intersect affine space $\mathbb{A}^{n}$ which is the covering of projective space . But $\mathbb{A}^{n}$ has dimension $n$, hence $X^0$ must has dimension $n$ because in affine integral closed scheme, all dense open subset has same dimension with the whole space. Thus, we can conclude that $X^0 \cong \mathbb{P}^{n}$ and thus also $X$. For the converse, $\varphi^{-1}(x) \cong \mathbb{P}^n$ and hence must be ismorphic to all $X^0$. Hence $T_0$ must be a point because if it has more than two, then $\varphi^{-1}(x_1) = \varphi^{-1}(x_2) = X^0$, which must not happen.
Conferences:
[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996.
Theorem: Suppose $C_x$ is a $d$- dimensional linear subspace of $\mathbb{P}(T_{x}X)$ for a general point $x \in X$. There is a dense open subset $X^0$ of $X$ and a $\mathbb{P}^{d+1}$-bundle $\varphi: X^0 \rightarrow T^0$ such that any curve from a minimal covering of rational curves, denote $H$, meeting $X^0$ is a line on a fibre of $\varphi$.
For the application of this theorem, I will write later. Now, I will give a part of the construction of
$X^0$ and $\varphi$, for the second statement, you can see theorem 3.1 in [Arau06].
First, there is some notion and properties that reader should know. I will copy the open part of [Arau06] right here.
Let $X$ be a smooth complex projective variety and, assume that X is uniruled. Let $H$ be an irreducible component of $RatCurves^{n}(X)$. We say that $H$ is a covering family if the corresponding universal family dominates $X$. A covering family $H$ of rational curves on $X$ is called minimal if, for a general point $x \in X$, the subfamily of $H$ parametrizing curves through $x$ is proper.
Let $x \in X$ be a general point and denote by $H_x$ the normalization of the subscheme of $H$ parametrizing rational curves passing through $x$. Let $\pi_x: U_x \rightarrow H_x$ and $\eta_x: U_x \rightarrow X$ be the universal family morphisms, so that $U_x$ is normal and $\pi_x$ is a $\mathbb{P}^1$-bundle ([Kol96, II.2.12]). Denote by $locus(H_x)$ the closure of the image of $\eta_x$ (with the reduced scheme structure). We remark that a rational curve smooth at $x$ is parametrized by at most one element of $H_x$.
For the concepts of $RatCurves^{n}(X)$ and $Univ^{rc}(x,X)$ ( $U_x$ here is $Univ^{rc}(x,X)$, readers can c.f. [Kol96, II.2].
Now, we will come to two key lemmas for constructing $X^0$ and $T^0$.
Remark: There is a result in [Kol96, II.2.15, II.3.5] state that $X$ is smooth over $locus(H_x)$. And we can also obtain $D$ as $D_x = \overline{C}_x \subset T_{x}X$ with $\overline{C}_x$ correspond to $C_x$ in $\mathbb{P}(T_{x}X)$.
Lemma 1: Suppose that $C_x$ is a union of linear subspaces of $\mathbb{P}(T_{x}X)$ for
general $x \in X$, with $X$ is uniruled projective variety and $D$ be as defined above. Then $D$ is tangent to $locus(H_x)$ for a general point $x \in X$. In particular, $locus(H_x)$ is smooth at $x$.
Lemma 2: Suppose that $C_x$ is a union of $d$-dimensional linear subspaces of $\mathbb{P}(T_{x}X)$ for general $x \in X$, with $X% is uniruled projective variety. Then, for a general point $x \in X$, the normalization of $locus(H_x)$ is isomorphic to $\mathbb{P}^{d+1}$. Under this isomorphism, the rational curves on $locus(H_x)$ parametrized by $H_x$ come from lines on $\mathbb{P}^{d+1}$ passing through a fixed point.
Now the first lemma helps us to provide a map from $locus(H_x)$ to the tangent point $x$, and from lemma 2, we can image that in $X^0$ we need $locus(H_x)$ must be its germ at $x$. From the "general" property, there must be a open dense subset $U$ s.t the condition that it contain all $locus(H_x)$ as general fibres and then provide a morphism to the point $x$ in the set of "tangent point" $T$. We see that $X^0$ and $T^0$ will be $U$ and $T$. And hence, we can come to a morphism $\varphi: X^0 \rightarrow T^0$ (in [Arau06], Araujo argued that $\varphi$ obtained by using "Frobenius' theorem", but I don't know what it is).
This constructing is useful for futher studies in the generalization of Hartshorne's conjecture and there is a lovely proposition that can play an important role in a proof of a even more general version of Hartshorne's conjecture.
Propostion: Let $C_x, X, X_0, T_0$ and $\varphi$ as above. Then $d = n-1$ and $X \cong \mathbb{P}^n$ if and only if $dim T_0 = 0$.
Proof of proposition. If $dim T_0 = 0$, then $T_0$ is a point. $\varphi^{-1}(T^0) = X^0 \cong \mathbb{P}^{d+1}$, but $X^0$ is dense in a projective variety $X$ with dimension $n$, then $X^0$ will intersect affine space $\mathbb{A}^{n}$ which is the covering of projective space . But $\mathbb{A}^{n}$ has dimension $n$, hence $X^0$ must has dimension $n$ because in affine integral closed scheme, all dense open subset has same dimension with the whole space. Thus, we can conclude that $X^0 \cong \mathbb{P}^{n}$ and thus also $X$. For the converse, $\varphi^{-1}(x) \cong \mathbb{P}^n$ and hence must be ismorphic to all $X^0$. Hence $T_0$ must be a point because if it has more than two, then $\varphi^{-1}(x_1) = \varphi^{-1}(x_2) = X^0$, which must not happen.
Conferences:
[Arau06] Carolina Araujo. Rational curves of
minimal degree and characterizations of projective
spaces. Math. Ann., 335(4): 937 -- 951, 2006.[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996.
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