A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{O}_{X}} $. Such a map $ \phi $ is called a splitting. A closed sub-scheme $ Y $ of $ X $ is compatibly split if there exists a splitting $ \phi $ such that $ \phi(F_{\ast}(\mathcal{I}_{Y})) \subseteq \mathcal{I}_{Y} $. Note that $ \mathcal{I}_{Y} \subseteq \phi(F_{\ast}(\mathcal{I}_{Y})) $ already.
If $ X $ is a normal variety, then $ X $ if $ F $-regular if for all effective Weil divisors $ D $ there is an $ e \in \mathbb{N} $ such that $ \mathcal{O}_{X} \to F^{e}_{\ast}(\mathcal{O}_{X}) \to F^{e}_{\ast}(\mathcal{O}_{X}(D)) $ splits.
If $ Z $ is a Mori Dream Space, then let $ \operatorname{cx}(Z) $ be the least natural number $ n \in \mathbb{N} $ such that the Cox ring of the blow-up of $ Z $ at $ n $-points in general position is not finitely generated.
If $ Z $ is an $ F $-regular, normal, projective variety, then let $ b(Z) $ be the least natural number $ n \in \mathbb{N} $ such that if $ W $ is the blow-up of $ Z $ at $ n $-points in general position, then $ -K_{W} $ is not big.
In https://mathoverflow.net/posts/439107/edit @KarlSchwede pointed out that if one keeps blowing up $ \mathbb{P}^{2}_{k} $ (which is $ F $-regular) at points in general position, then eventually one obtains a variety which is not $ F $-regular. We know that by the time we have blown-up $ 9 $ points that $ -K_{W} $ is no longer big. If $ W_{n} $ is the blow-up of $ \mathbb{P}^{2}_{k} $ at $ n $-points in general position, then I do not know if $ -K_{W_{n}} $ is still big for $ n<9 $ as I don't know if $ H^{1}(W_{n},-mK_{W_{n}}) $ vanishes for $ n<9 $ and $ m>>0 $. If it does, then $ \operatorname{vol}(-K_{W_{n}})>0 $ for $ n<9 $ and $ b(\mathbb{P}^{2}_{k}) $ is exactly nine.
Needless to say, $ \operatorname{cx}(\mathbb{P}^{2}_{k}) $ is exactly nine by a result of Mukai.
If one looks at Some Non-Finitely Generated Cox Rings Jose Luiz Gonzalez and Kalle Karu, then in that paper Gonzales and Karu examine when the Cox ring of the blow-up of a weighted projective space at exactly one smooth point is not finitely generated. On page 5 of that paper they give a few examples of weights $ (a:b:c) $ for which the Cox ring of $ \operatorname{Bl}_{e}(\mathbb{P}(a:b:c)) $ is not finitely generated.
Examples are $ (7:15:26), (7:17:29), (7:22:17) $. A toric Bezout's theorem for weighted projective spaces (see A Toric Proof of Bezout's Theorem for Weighted Projective Spaces By Berndt Ivar Utstol Nodland) states that if $ E_{1},\dots,E_{n} $ are $ n $ torus invariant divisors on $ \mathbb{P}(q_{0}:\cdots:q_{n}) $, then $ E_{1}\cdots E_{n} = \left(\prod_{i=1}^{n} \operatorname{deg}(E_{i})\right)/(\prod_{j=0}^{n} q_{j}) $. The anticanonical divisor of a weighted projective surface is $ -\sum_{i=1}^{3} D_{\rho_{i}} $. Note that for all of these weighted projective spaces $ Z $ that $ (-K_{Z})^{2}<1 $. As a result $ b(Z)=\operatorname{cx}(Z) $ for these weighted projective spaces.
Is it worth looking into this pattern, or does anyone know an example of an $ F $-regular, Mori dream space $ Z $ such that $ b(Z) \ne \operatorname{cx}(Z) $?
