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The linear system of $|-K_{S_d}|$ can be identified with a multiple of the linear system of hyperplane sections.

A general member of this will actually miss the vertex, and it is smooth everywhere.

Those elements that contain the vertex are identified with hyperplane sections through the vertex. In turn, these hyperplanes are themselves cones over hyperplanes in $\mathbb P^d$. In other words a general member containing the vertex is isomorphic to $d+2$ lines in general position in $\mathbb P^{d+1}$ going through a fixed point (the vertex).

The linear system of $|-K_{S_d}|$ can be identified with a multiple of the linear system of hyperplane sections.

A general member of this will actually miss the vertex, and it is smooth everywhere.

Those elements that contain the vertex are identified with hyperplane sections through the vertex. In turn, these hyperplanes are themselves cones over hyperplanes in $\mathbb P^d$. In other words a general member containing the vertex is isomorphic to $d+2$ lines in general position in $\mathbb P^{d+1}$ going through a fixed point (the vertex).

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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 general position in $\mathbb A^{d+2}$ going through a fixed point.

The linear system of $|-K_{S_d}|$ can be identified with a multiple of the linear system of hyperplane sections.

A general member of this will actually miss the vertex, and it is smooth everywhere.

Those elements that contain the vertex are identified with hyperplane sections through the vertex. In turn, these hyperplanes are themselves cones over hyperplanes in $\mathbb P^d$. In other words a general member containing the vertex is isomorphic to $d+2$ lines in general position in $\mathbb P^{d+1}$ going through a fixed point (the vertex).