What is the minimal $ n $ such that the Cox ring of the blow up of a simplicial, $ r $-dimensional, toric variety at $ n $ points in g.p. is not f.g.? In Shigeru Mukai's paper "Counterexample to Hilbert's 14th Problem for the 3-dimensional Additive Group," Mukai proved that if $ \frac{1}{r+1}+\frac{1}{n-r-1} \le \frac{1}{2} $, then the blow up of $ \mathbb{P}^{r} $ at $ n $ points in general position is not finitely generated.  Is there a lower bound on $ n $ so that if $ X_{\Sigma} $ is a $ r $-dimensional, projective, simplicial toric variety over a field of arbitrary characteristic, then the Cox ring of the blow up of $ X_{\Sigma} $ at $ n $ points in general position is not finitely generated?
@user347489 reminded me that I should clarify that I do not merely want a number $ n $ such that for some $ r $-dimensional toric variety $ X_{\Sigma} $ the blow-up of $ n $-points in general position has a non-finitely generated Cox ring, but an $ n(r) $ such that for all toric varieties $ X_{\Sigma} $ the Cox ring of the blow-up of $ X_{\Sigma} $ in $ n(r) $-points in general position is not finitely generated.  This question may be too broad or not have enough parameters.  If one specifies also that the class group of $ X_{\Sigma} $ is free of rank $ \ell $, does that add any information?
 A: There is an ongoing effort to understand this problem, since it's already very complicated for the simplest examples: weighted projective planes blown up at a single general point. Below I mostly discuss this case.
Historically, the first geometric study of the problem was done by Cutkosky in '91, motivated by an equivalent problem originating from commutative algebra. The first non-MDS examples were found in '94 by Goto, Nishida and Watanabe, and, as far as I know, the area became mostly inactive after this (with the exception of some works by Kurano and different collaborators, see below). Twenty years later Castravet and Tevelev's proof that $\overline{M}_{0,n}$ is not a MDS uses these blow-ups in an essential way. The question you ask has received renewed interest since then.
Gonzalez-Anaya, Gonzalez and Karu have a series of papers addressing this problem. See for example this one, where they offer a summary of their previous work and exhibit some deeper structures underlying the problem. Their last preprint finds examples of projective toric surfaces with Picard number 1 with a semiopen effective cone, see here.
Other papers addressing the problem include work of Kurano with several coauthors like Cutkosky, Matsuoka and Nishida (in his latest paper with Inagawa they settle a conjecture of He), work of Laface and Ugaglia, etc.
It's worth mentioning that not too long ago Castravet, Laface, Tevelev and Ugaglia showed that $\overline{M}_{0,n}$ has a non-polyhedral pseudoeffective cone for $n\geq 10$ by showing the same property for a toric surface with higher Picard number.
I can offer some more details if you wish.
