A quadric hypersurface (over an algebraically closed field of characteristic zero) in $\mathbb{P}^n$ for $1\leq n\leq 3$ is a toric variety. (Namely, it's isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$, $\mathbb{P}^1$, or two points, depending on $n$.)

Is it true for general $n$ that quadric hypersurfaces are toric varieties? Again, over an algebraically closed field of characteristic zero.

According to Harris' book (**Algebraic Geometry A First Course**), on page 288, the blowup of a smooth quadric in $\mathbb{P}^n$ at a smooth point is isomorphic to the blowup of $\mathbb{P}^{n-1}$ along a smooth quadric $C$ of dimension $n-3$, but I'm not sure if that helps. And I'm assuming that the smooth case is the only interesting one, since singular quadrics are just cones over smooth ones, so I'd be content with an answer just about the smooth quadrics.

smoothtoric with Picard rank $1$ is $\mathbb P^n$) $\endgroup$ – user47305 Feb 19 '16 at 16:54