Mori's “projective manifolds with ample tangent bundles”, Theorem 4

See http://www.jstor.org/stable/1971241?seq=6#page_scan_tab_contents, pages 598 and 599.

Let $X$ be a nonsingular projective variety, $\alpha : D \rightarrow H := \text{Hom}_k(\mathbb{P}^1, X; j)$ where

• $D$ is non-singular
• $[\phi] \in \alpha(D)$ where $\phi:\mathbb{P}^1\rightarrow C \subset X$
• $\phi(0) = P, \phi(\infty)=Q \in X$
• $j := \phi|_{0,\infty}$

(We do know that dim$_{[\phi]}H \geq 2$.) $\alpha$ induces a morphism $F:\mathbb{P}^1\times D \rightarrow X\times D$. Let $\bar D$ be the compactification of $D$, $Y$ be the closure of the image of $F$, and $\tilde Y$ be the normalization of $Y$. This gives a morphism

$$\pi: \tilde Y \rightarrow Y \rightarrow X\times \bar D \rightarrow \bar D$$

In Theorem 4 of the paper, Mori claims that to show $$\pi^{-1}(D) \cong \mathbb{P}^1 \times D,$$ it is enough to show that "$F|_U$ is an immersion for some open set $U$ of $\mathbb{P}^1$ because $F$ is finite."

The remainder of the paragraph then goes on to prove this, but I do not yet understand this claim. Anyone?

Suppose $F\rvert_{U}\colon U \to X \times D$ is an immersion for some open subset $U \subseteq \mathbf{P}^1 \times D$, and denote the image of $\mathbf{P}^1 \times D$ in $X \times D$ by $W$. Then, since $\mathbf{P}^1 \times D$ is normal, the morphism $F$ factors through the normalization $\widetilde{W} \to W$ (see this MathOverflow answer). Now this morphism $\mathbf{P}^1 \times D \to \widetilde{W}$ is finite and birational, and $\widetilde{W}$ is normal, hence $\mathbf{P}^1 \times D \to \widetilde{W}$ is an isomorphism by [Stacks, Tag 0A81]. Thus, denoting $$\require{AMScd} \begin{CD} \widetilde{W} @>\subset>> \widetilde{Y}\\ @VVV @VVV\\ W @>\subset>> Y\\ @VVV @VVV\\ X \times D @>\subset>> X \times \overline{D}\\ @VVV @VVV\\ D @>\subset>> \overline{D} \end{CD}$$ we see that $\pi^{-1}(D) \cong \mathbf{P}^1 \times D$.
• If we take $\tilde W$ to be $\tilde Y$, then indeed we have a morphism from $\mathbb{P}^1 \times D$, which is still finite and birational. Note here that we can factor the immersion as an open immersion followed by a closed immersion because $F|_U$ is finite, hence F is birational. (stacks.math.columbia.edu/tag/01QV) – numberjedi May 14 '17 at 15:47
• @numberjedi I think the issue was the word "its", which I have now replaced with something clearer. If we take W to be the image of $\mathbf{P}^1 \times D$, then the factorization works as you say. – Takumi Murayama May 14 '17 at 15:53