The context of this question is Hartshorne, Chapter II, Theorem 7.6. Let $X$ be a scheme of finite type over a noetherian ring $A$, and $\mathscr{L}$ an invertible sheaf on $X$. The claim is that $\mathscr{L}$ is ample if and only if $\mathscr{L}^{\otimes m}$ is very ample over $\text{Spec }A$ for some $m > 0$.
For the forward direction, suppose $\mathscr{L}^{\otimes m}$ is very ample. Then there is an immersion $i X \rightarrow \mathbb{P}_{A}^{n}$. Hartshorne then says to let $\overline{X}$ be the closure of $X$ in $\mathbb{P}_{A}^{n}$.
What does this mean? Is this referring to the set theoretic closure, with the reduced induced subscheme structure? Or is it referring to the scheme theoretic closure, or something else?