How to show that cubic surface is not toric?

Question

I know how to show $\mathbb{P}^2$ blowing up at 0,1,2,3 points are toric, but I don't know how to show that blowing up at 4-8 points are not toric. can someone show that blowing up $\mathbb{P}^2$ at 6 points (which is a cubic surface in $\mathbb{P}^3$) is not toric to inspire me?

Answer 1

The technique I'm imagining for showing this would be exactly the same for blowing up any $n>3$ points in $\mathbb{P}^2$. Here's a hint- I'd be glad to fill in the details a little if you so desire.

Hint: Look at the torus action on the blowup and compare it with the induced action on $\mathbb{P}^2$ via the blow-down map. What can you say? Are there any special points?

Any automorphism of the blowup must send the $-1$ curves to $-1$ curves, which means the points in $\mathbb{P}^2$ that are the images of the $-1$ curves are fixed by the torus action. But the torus action fixes only 3 points of $\mathbb{P}^2$, so the number of exceptional curves in the blowup must be at most 3.