How does one determine the singular points of a toric variety?

Question

Consider the toric surface corresponding to the fan $\Delta$ consisting of $$\sigma_1=\langle e_1,-e_1+2e_2\rangle$$ $$\sigma_2=\langle-e_1+2e_2,-e_1-2e_2\rangle$$ $$\sigma_3=\langle -e_1-2_2,e_1\rangle$$ and their faces.

Since the fan is simplicial, the toric variety is a simplicial toric variety. So it has singularities. However I am unable to identify what the singular points are.

For example if I just take the cone $\sigma_1$, the corresponding affine toric variety has a singularity at the origin. But how do I find the singular points of the toric variety corresponding to $\Delta$?

Thank you.

Answer 1

Affine varieties $\mathbf A_{\sigma_1}, \mathbf A_{\sigma_2}, \mathbf A_{\sigma_3}$ form an affine open covering of $\mathbb P_{\Sigma}$, hence, to find singular points of $\mathbb P_{\Sigma}$ it is enough to find singular points of $\mathbf A_{\sigma_1}, \mathbf A_{\sigma_2}, \mathbf A_{\sigma_3}$. Toric varieties are normal, so singular locus is codimension $2$ and dimension $0$ in your case. Every singular point is a stable point of torus action in 2-dimensional case, if it is not we can apply torus action to generate at least 1-dimensional line of singular points. So, the origins of $\mathbf A_{\sigma_1}, \mathbf A_{\sigma_2}$ and $\mathbf A_{\sigma_3}$ are only singular points of $\mathbb P_{\Sigma}$, since all cones $\sigma_1,\sigma_2,\sigma_3$ are not smooth. All 3 points are pairwaisly distinct since we have cones-orbits 1-1 corresponence.