The Stacks project

Proof. For projective space one can show directly that the field $k(x_1, \ldots , x_ r)$ is $C_ r$ (exercise).

General case. Without loss of generality, we may assume $X$ to be projective. Let $f \in k(X)[T_1, \ldots , T_ n]_ d$ with $0 < d^ r < n$. Say the coefficients of $f$ are in $\Gamma (X, \mathcal{O}_ X(H))$ for some ample $H \subset X$. Let $\mathbf{\alpha } = (\alpha _1, \ldots , \alpha _ n)$ with $\alpha _ i \in \Gamma (X, \mathcal{O}_ X(eH))$. Then $f(\mathbf{\alpha }) \in \Gamma (X, \mathcal{O}_ X((de + 1)H))$. Consider the system of equations $f(\mathbf{\alpha }) =0$. Then by asymptotic Riemann-Roch (Varieties, Proposition 33.45.13) there exists a $c > 0$ such that

• the number of variables is $n\dim _ k \Gamma (X, \mathcal{O}_ X(eH)) \sim n e^ r c$, and

• the number of equations is $\dim _ k \Gamma (X, \mathcal{O}_ X((de + 1)H)) \sim (de + 1)^ r c$.

Since $n > d^ r$, there are more variables than equations. The equations are homogeneous hence there is a solution by Lemma 59.67.7. $\square$