# An intersection-theoretic criterion for Moishezon surfaces by Sakai

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?

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.