Let $D \in S$ be an ample class.

If $D^2 =2$ and there does not exist a class $F\in S$ such that

$F^2 = 0$ and $F\cdot D = 1$, then $$\varphi_D : X \longrightarrow \mathbb{P}^2$$ is a **double cover**.

If $D^2 = 4$ and there does not exist a class $F\in S$ such that

$F^2 = 0$ and $F\cdot D \in \{1,2\}$ then $$\varphi_D : X \longrightarrow \mathbb{P}^3$$ embeds $X$ as a **quartic surface**.