Let $k$ be an algebraically closed field of characteristic zero and let $X$
be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with an open orbit.


Now comes the question: If $X$ is quasi-affine and smooth, is the $k$-algebra $k[X]$ of regular functions on $X$ always finitely generated (over $k$)? Is this true for arbitrary toric $X$?


I know, that in general (i.e. when $X$ is not necessarily toric), then $k[X]$ can be non-finitely generated, see e.g. the article of Winkelmann [Win03].


Any proof, counter-example or reference to my question would be very welcome!


[Win03] *Winkelmann, J\"org*, [**Invariant rings and quasiaffine quotients**](http://dx.doi.org/10.1007/s00209-002-0484-9), Math. Z. 244, No.1, 163-174 (2003). [ZBL1019.13003](https://zbmath.org/?q=an:1019.13003).



The algebra of regular functions of a quasi-affine toric variety
One can show that for any normal spherical variety $X$ the algebra $k[X]$ is finitely generated. No quasi-affinity or smoothness needed. This holds more generally for $G$-varieties ($G$ connected reductive) where $G$ has an open orbit and a Borel subgroup has an orbit of codimension $\le1$. The latter condition is optimal. See the paper [Knop, F: Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Mathematische Zeitschrift 213 (1993) 33-35](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002441497).


