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It is known that a rational elliptic surface is a blow-up of $\mathbb{P}^2$ at 9 points. More precisely it is obtained as the blow-up of the base locus of a pencil of cubic curves in $\mathbb{P}^2$. The question is, whether or not we can choose such a pencil (at least the base locus should be in the torus fixed points).

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    $\begingroup$ If you are asking whether the blow up of the plane at $9$ points that are base locus of a pencil of cubics can be a toric surface, then the answer is no. In fact, such an elliptic surface contains infinitely many $(-1)$-curves, so in particular its Cox ring is not finitely generated. On the other hand, any toric variety has a finitely generated Cox ring. See also mathoverflow.net/questions/72917/… $\endgroup$
    – Francesco Polizzi
    Commented May 21, 2016 at 8:37

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