7 – Proj | PModChecker 2/2

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

If $D^2=6$ and there does not exist a class $F\in S$ such that
$F^2=0$ and $F\cdot D\in \{1,2\}$, then $${\phi }_D:X⟶{\mathbb{P}}^4$$ embeds $X$ as a degree $6$ surface.

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If $D^2=8$ and there does not exist a class $F\in S$ such that
$F^2=0$ and $F\cdot D\in \{1,2,3\}$ then $${\phi }_D:X⟶{\mathbb{P}}^5$$ either embeds $X$ as a generically transverse intersection of three quadrics with only rational double points, or realizes $X$ as a double cover of a Veronese surface.

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