Let $\mathbb{P}^1= C_d \subset \mathbb{P}^d$ be the rational normal curve of degree $d$ and $S_d \subset \mathbb{P}^{d+1}$ be the projective cone over $C_d$. $S_d$ is a typical example of a log del Pezzo surface.

I want to know about the singularity of general elements $D_d$ of the linear system $|-K_{S_d}|$. I think $D_d$ is singular at the vertex. Does $D_d$ have only a nodal singularity at the vertex?