Singularity of a surface in $\mathbb{P}(1,1,1,a)$ passing through the vertex

Question

I would like to understand better the kind of singularities that we obtain in hypersurfaces in weighted projective spaces. For instance, let us consider the surface $S:x_0^{2a}+x_1^{2a}+x_2^{2a}=0$ of $\mathbb{P}(1,1,1,a)$ with coordinates $(x_0,x_1,x_2,y)$.

If I am not wrong, locally around the vertex $(0:0:0:1)$ this corresponds to the variety $z_0^{2a}+z_1^{2a}+z_2^{2a}=0$ in $\mathbb{A}^3/\mathbb{Z}_a$. Here the action is given by, $$ (\epsilon, (z_0,z_1,z_2))\mapsto (\epsilon z_0,\epsilon z_1,\epsilon z_2) $$ where $\epsilon$ is a primitive $a$-root of unity.

So, my question is: what can we say about $(0:0:0:1)\in S$? What kind of singularity do we have there?

Answer 1

Note that $\mathbb{P}(1,1,1,a)$ is the cone over the $a$-th Veronese embedding of $\mathbb{P}^2$. Consequently, the surface $S$ is the cone over the plane Fermat curve $C$ of degree $2a$ embedded via the complete linear system $\mathcal{O}_{\mathbb{P}^2}(a)\vert_C$.