# Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$

In the remark on the bottom of page 5 of this paper, the author states that

It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In fact, by the Iitaka fibration, there exists a variety $Y$ of dimension $\kappa(X, D)$, and divisor $D'$ on $Y$ such that $R(X,D) \cong R(Y,D')$.

Assume $X$ to be a complete normal algebraic variety.

Can anyone give a reference for this well-known fact, or elucidate the "$\cong$" in his brief explanation? Of particular interest to me is the case with $X$ a surface and $\kappa(D)=1$.

• I recommend that you contact the authors of that paper. Anyway, it appears that the isomorphism, interpreted strictly, is false. There are Halphen surfaces, cf. the excellent answer of Polizzi and comment of Loughran to the following MO question: mathoverflow.net/questions/283609/… In this case, the divisor class $D=-K_X$ has the property that for some integer $m>0$, the subring $R(X,mD)$ is isomorphic to $R(\mathbb{P}^1,\mathcal{O}(1))$. Since $m$ is strictly larger than $1$, $R(X,D)$ is not $\cong R(Y,D')$. – Jason Starr Dec 4 '17 at 11:01
• If you read the introduction to that paper, I believe that the authors are implicitly allowing to replace the divisor $D$ by a positive integer multiple before constructing an isomorphism with $R(Y,D')$. That would explain the claim that there always exists a curve $Y$ and an ample divisor $D'$ such that $R(X,D)\cong R(Y,D')$. – Jason Starr Dec 4 '17 at 11:07
• For every linear system $(\mathcal{L},V)$ on $X$, there exists a normal projective variety $Y$, there exists an ample invertible sheaf $\mathcal{A}$ on $Y$, and there exists a finite morphism from a normal variety, $\nu:\widetilde{X}\to X\times Y$, such that (i) the projection $p=\text{pr}_X\circ \nu:\widetilde{X}\to X$ is projective and birational, (ii) the morphism $q=\text{pr}_Y\circ \nu:\widetilde{X}\to Y$ is a contraction, and (iii) there is a morphism $\phi:q^*\mathcal{A}\to p^*\mathcal{L}$ such that $p^*V$ is contained in the image of $H^0(Y,\mathcal{A})$ . . . – Jason Starr Dec 4 '17 at 13:35
• . . . For a fixed $\mathcal{M}$ on $X$, let $(\mathcal{M}^{\otimes m},V),$ $m\in \mathbb{Z}_{>0}$, be such that $Y$ has maximal dimension. For every $n\in \mathbb{Z}_{>0}$, for every $s\in H^0(X,\mathcal{M}^{\otimes n})$, for the linear system $(\mathcal{M}^{\otimes mn},V')$ spanned by $V^n$ and $s^m$, for the associated contraction $\nu':\widetilde{X}'\to Y'$, there is a unique contraction $r:Y'\dashrightarrow Y$ and $\psi:r^*\mathcal{A}^{\otimes n} \to \mathcal{A}'$ such that $\phi^n$ factors through $\psi$. If $Y$ and $Y'$ have dimension $1$, then $r$ is an isomorphism. – Jason Starr Dec 4 '17 at 13:42
• The point of the second post is the final line. Every contraction between normal, projective curves is automatically an isomorphism. If that confuses you, I advise you to read about the valuative criterion of properness and Zariski's Main Theorem. – Jason Starr Dec 4 '17 at 14:40