An intersection-theoretic criterion for Moishezon surfaces by Sakai

Question

I have found myself in a situation where I need to show that a particular normal analytic surface is (projective) algebraic. I am trying to use a result by Brenton's 1977 paper Some Algebraicity Criteria for Singular Surfaces which states that a normal compact surface is algebraic if it is Moishezon with geometric genus $0$.

In Sakai's 1984 paper Weil Divisors on Normal Surfaces, he mentions that a normal surface is Moishezon if and only if it has a divisor with positive self-intersection but he gives no reference for this fact.

How does having a divisor with positive self-intersection mean a surface if Moishezon (meaning the transcendence degree over $\mathbb{C}$ of its meromorphic function field is $2$) and/or where can I find a reference for this fact?

Answer 1

Essentially, if there is a divisor with positive self-intersection on a surface, then there also is one with positive volume, i.e., a big divisor. And, clearly, having a big divisor is equivalent to being Moishezon. But it's easier to pass to a resolution first, to avoid dealing with non-Cartier divisors:

Passing to a resolution of singularities, the pull-back of said divisor will be a Cartier divisor with (the same) positive self-intersection. If this resolution is Moishezon, then so is the one we started with. Therefore, we may suppose that the surface is smooth and the divisor is Cartier.

Now let $C$ be a Cartier divisor on a smooth compact complex surface $X$ satisfying $C^2>0$. By (Asymptotic) Riemann-Roch, $$h^0(nC)+h^0(K_X-nC)\geq \chi(\mathcal O_X(nC)) = \tfrac12 C^2 n^2+O(n)$$ grows quadratically in $n^2$ as $|n|\to\infty$.

Therefore, after replacing $C$ with either $nC$ or $K_X-nC$ for some sufficiently large $n$, we may assume that $C$ is effective (and still has positive self-intersection). Repeating the asymptotical argument above, noting that then $h^0(K_X-nC)$ is non-increasing with $n$, hence bounded, it follows that $h^0(nC)\geq \tfrac12 C^2 n^2+O(n)$ as $n\to\infty$. In particular, $C$ is big and $X$ is Moishezon.