I am reading the paper "Mori Dream Spaces and GIT" by Hu and Keel.


<https://arxiv.org/abs/math/0004017>


I cannot understand the proof of Lemma 1.6 in it.


Let $X$ be a normal projective variety.
Assume that $D$ is a divisor on $X$ such that $R(X,D) = \bigoplus\_{m \in \mathbb{Z}\_{\ge0} } H^0(X,\mathcal{O}\_X(mD))$ is finitely generated.


Lemma 1.6 states that the natural rational map $\varphi\_D \colon X \dashrightarrow Y := Proj(R(X,D))$ is a rational contraction, i.e, there exits a resolution $\Phi \colon \tilde{X} \to Y$ of $\varphi\_D$ such that $\Phi\_\*(\mathcal{O}\_{\tilde{X}} (E)) \simeq \mathcal{O}\_Y$ for any effective exceptional divisor $E$.
Moreover, $D$ can be written as $ D = \varphi\_D^\*(A) + F $ for an ample divisor $A$ on $Y$ and a $\varphi\_D$-fixed divisor $F$.


Outline of the proof of the paper is as follows:


(1) The finite generation of $R(X,D)$ implies that $D$ can be written as $D = M+F$ for a movable divisor $M$ and an effective divisor $F$ such that
$Sym\_k(H^0(X,M)) \to H^0(X,kM)$ is surjective and $H^0(X,kM) \to H^0(X,kM +rF)$ is an isomorphism for any $k>0$ and $r>0$ after replacing $D$ by its multiple.


(2) After taking appropriate resolution, we may assume that $\varphi\_D$ is a regular
and $M = \varphi\_D^\* A $. Now we can see $F$ is $\varphi\_D$-fixed by the isoromorphism $H^0(X,kM) \simeq H^0(X,kM +rF)$.


Here is my question :


[1] How can wee see that $\varphi\_D$ is a rational contraction? Is it a well-known result?


[2] Why does the isomorphism $H^0(X,kM) \simeq H^0(X,kM +rF)$ implies that
$F$ is $\varphi\_D$-fixed?


Can anyone who understands these results answer my questions?



Rational contraction and Proj of section ring
Let's assume that $|kD|=|kM|+kF$ where $|M|$ is base point free and moreover $Sym ^kH^0(M)\to H^0(kM)$ is surjective for any $k>0$. This can be achieved replacing $X$ by an appropriate resolution and $k$ by a multiple. Let $f:X\to Y$ be the morphism induced by $|M|$ so that $f^\*A=M$.


For [1], we wish to show that $f\_\* \mathcal O \_X=\mathcal O \_Y$. The Stein factorization $X\to Z\to Y$ is defined by ${\rm Spec} f\_\* \mathcal O \_X$. We have $H^0(X,kM)=H^0(f\_\*(kM))=H^0(f\_\* \mathcal O \_X\otimes \mathcal O \_Y(kA))$ and so $|kM|$ in fact defines the morphism $X\to Y$. Since $Sym ^kH^0(M)\to H^0(kM)$ is surjective, then $|M|$ defines the same morphism.


For [2], Since $H^0(X,kM)=H^0(X,kM+rF)$, then
$H^0(\mathcal O \_Y(kA))\cong H^0(\mathcal O \_Y(kA)\otimes f\_\* \mathcal O \_X(rF))$. For $k\gg r$, the sheaf $\mathcal O \_Y(kA)\otimes f\_\* \mathcal O \_X(rF)$ is globally generated and so $f\_\* \mathcal O \_X(rF)\cong \mathcal O \_Y$. (See also Lemma 3.2 <https://arxiv.org/pdf/1104.4981.pdf>).


Hope this helps.


