What does it mean for a line bundle to be $n-$very ample?

Question

I know what a very ample line bundle is, but cannot find a definition of an $n-$very ample line bundle, where $n$ is some integer

Answer 1

Here is one notion that goes by the name "$k$-very ample":

Definition [Beltrametti–Sommese 1991]. Let $L$ be a line bundle on a variety $X$. We say that $L$ is $k$-very ample for $k \ge 0$ if for every zero-dimensional subscheme $Z$ of length $k+1$, the restriction map $$H^0(X,L) \longrightarrow H^0(X,L \otimes \mathcal{O}_Z)$$ is surjective.

The intuition is that if $L$ is $k$-very ample for $k > 1$, then it induces a higher-order embedding of $X$. You might imagine that there could be other ways to capture this notion, and indeed, there are many higher-order ampleness conditions you can put on a line bundle; see this MathOverflow answer for a summary. The first place where such ideas appeared is probably [Beltrametti–Francia–Sommese 1989].

Some properties you might expect about $k$-very ampleness are rather difficult to prove: for example, it is non-trivial that the tensor product of a $k$-very ample line bundle and an $l$-very ample line bundle is $k+l$-very ample [Hinohara–Takahashi–Terakawa 2004].