TITLE: What are the general techniques for proving a variety is not toric?
QUESTION [9 upvotes]: When I read the paper "A survey on Cox rings" with the link below:
http://math.berkeley.edu/~velasco/Survey.pdf
In Section 4.2 it is mentioned that the Del Pezzo surface given by blowup of $\mathbb{P}^2$ at $s>3$ general points is not toric. Is there an easy way to see this?

REPLY [2 votes]: To be a toric variety implies many other things. For instance if $X$ is a toric variety, then

$X$ is a Mori Dream Space, i.e. $Cox(X)$ is finitely generated. Indeed $X$ is a toric variety if and only if $X$ is a Mori Dream Space and $Cox(X)$ is a polynomial ring.
$X$ is log Fano, in particular $-K_X$ is big and movable.
Any nef divisor is semiample.

Therefore to show that $X$ is not toric one could prove that $-K_X$ is not big or work out a nef divisor which is not semiample.