Toric blowup at a torus-fixed point

If a smooth maximal cone is generated by a lattice basis $e_1,\dots ,e_n$, its star subdivision through $e_1+\cdots +e_n$ is the fan of the blowup at the corresponding torus-fixed point.

For an ample invariant divisor $D$ and the blowup $\pi$ of a torus-fixed smooth point in dimension at least two, $c{\pi }^{\ast }D-E$ is ample for sufficiently large integers $c$. In the subdivided cone, its local characters are $c{m}_{\sigma }+u_i$, where $u_i$ is dual to the omitted ray generator. The new strict inequalities have gap one on the omitted ray and grow linearly with $c$ elsewhere, proving the claim by the toric ampleness criterion.

The blowup of ${\mathbb{A}}^n$ at the origin is the total space of the tautological line bundle ${\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1)$. Its exceptional divisor is the zero section ${\mathbb{P}}^{n-1}$, and projection to the direction of a line gives the bundle map to ${\mathbb{P}}^{n-1}$.