If I am not mistaken, the rational map induced by $|-K_{S_d}|$ is the "obvious" one mapping $S_d$ to $C_d$. Then every member of that linear system consists of $d+2$ rays of the cone. In other words $D_d$ is isomorphic to $d+2$ lines in $\mathbb A^{d+2}$ going through a fixed point but so that no $3$ of them lie on the same plane.
Sándor Kovács
- 42.9k
- 2
- 109
- 155