Is there an example of a smooth projective variety $X$ (say, over $\mathbb{C}$) such that the $\mathbb{C}$-algebra $\bigoplus\_{n\geq 0}H^0(X,K\_X^{-n}) $ is not finitely generated?



Can $\bigoplus_{n\geq 0}H^0(X,K_X^{-n}) $ fail to be finitely generated?
Chenyang Xu claims that $X = \mathbb{P}\_S\left(\mathcal{O}\_S \oplus \mathcal{O}\_S(H) \right)$, where $S$ the blow-up of $\mathbb{P}^2$ along $9$ general points and $H$ is any ample Cartier divisor on $S$, provides such an example. See example 3.8 in his paper [K-stability for varieties with a big anti-canonical class](https://arxiv.org/pdf/2210.16631.pdf).


