How should I think about very ample sheaves?

Question

Definition. [Hartshorne] If $X$ is any scheme over $Y$, an invertible sheaf $\mathcal{L}$ is very ample relative to $Y$, if there is an imersion $i\colon X \to \mathbb{P}_Y^r$ for some $r$ such that $i^\ast(\mathcal{O}(1)) \simeq \mathcal{L}$.

My question is: what is the right way (interpret "right way" as you wish) to think about very ample sheaves? In particular, why is the word "ample" being used? What is it that I have an ample amount of? Degree 1 elements?

In the simple case when $Y=\text{Spec}(A)$ is affine then $i^\ast(\mathcal{O}(1))$ is just $\mathcal{O}(1)$ as defined on $\text{Proj} A[x_0,\ldots x_r]$, that is, it's the sheafification of the degree 1 part of the polynomial ring $A[x_0,\ldots,x_n]$. So it seems like the more general definition is just meant to generalize this phenomenon. Is this true? If so, why is it worth generalizing? What's special about degree 1 elements? The only thing I can think of is that the polynomial ring is generated as an $A$-algebra by its degree 1 elements.

As you can tell, my question is not very well formed, so feel free to add anything you think is relevant. I am also happy to expand on anything I've written here.

Answer 1

$\newcommand{P}{\mathbb{P}}\newcommand{E}{\mathscr{E}}\newcommand{O}{\mathscr{O}}\newcommand{L}{\mathscr{L}}$Let me first briefly summarize a construction from EGA (II, 4.2). Let $X$ and $Y$ be schemes, $q : X \to Y$ a morphism, $\E$ a quasi-coherent $\O_Y$-module, and $\P(\E)$ the projective bundle defined by $\E$. There is a bijection between the $Y$-morphisms from $X$ to $\P(\E)$ and equivalence classes of pairs $(\L, \varphi)$ of invertible $\O_X$-modules $\L$ and surjective homomorphisms $\varphi : q^*(\E) \to \L$, under the relation where $(\L, \varphi)$ and $(\L', \varphi')$ are identified if there is an isomorphism $\tau : \L \stackrel{\sim}{\to} \L'$ such that $\varphi' = \tau \circ \varphi$. This correspondence is such that a morphism $r : X \to \P(\E)$ corresponds to the invertible $\O_X$-module $r^*(\O_{\P(\E)}(1))$; see EGA for the details. In the particular case $\E = \O_Y^{n+1}$, $P = \P(\O_Y^{n+1}) = \P^n_Y$, it follows that morphisms $r : X \to \P^n_Y$ are in bijection with invertible $\O_X$-modules $\L$ and surjective homomorphisms $\varphi : q^*(\O_Y^{n+1}) \to \L$; but $q^*(\O_Y^{n+1}) = \O_X^{n+1}$ and surjective homomorphisms $\O_X^{n+1} \to \L$ are in bijection with surjective homomorphisms $\O_{X,x}^{n+1} \to \L_x \stackrel{\sim}{\to} \O_{X,x}$, which are again in bijection with tuples $(s_0, \ldots, s_n)$ of global sections of $\L$ that generate $\L$ (i.e. have no common zeros).

Now an invertible $\O_X$-module $\L$ is called very ample for $q$ if there exists a quasi-coherent $\O_Y$-module $\E$ and an immersion $i : X \to P = \P(\E)$ such that $\L$ is isomorphic to $i^*(\O_P(1))$. One immediately sees that this is equivalent to the condition that there exists a quasi-coherent $\O_Y$-module $\E$ and a surjective homomorphism $\varphi : q^*(\E) \to \L$ such that the corresponding morphism $X \to P = \P(\E)$ is an immersion. As we saw above, in the case $\E = \O_Y^{n+1}$, this means that $\L$ is globally generated by $n+1$ sections. Basically, the term very ample is referring to the global sections: roughly speaking, $\L$ is very ample if there are "enough" global sections to define an immersion into projective space.

Answer 2

Intuitively and for my answer, I am working over an algebraically closed field $k$.

I personally think about invertible sheaves as sheaves of functions on a scheme, where $s\in\mathcal L(U)$ can be evaluated at a point $P\in U$ in the sense that $s(P)$ is the image of $s$ under the morphism $\mathcal L(U) \to \mathcal L_P \cong \mathcal O_{X,P} \twoheadrightarrow \mathcal O_{X,P}/\mathfrak m_P=k(P)=k$. Of course, this depends on the local isomorphism we choose, but if I choose the same local isomorphism for $s_0,\ldots,s_n\in\mathcal L(U)$, then $[s_0(P):\ldots:s_n(P)]\in\mathbb P_k^n$ is well-defined as a point in projective space.

Hence intuitively, $\mathcal L$ is very ample if these functions can serve as coordinates, i.e. I have enough functions to distinguish points - and, in fact, this is close to an alternative characterization, see Proposition II.7.3 in Hartshorne (page 152): The morphism $\varphi:X\to\mathbb P^n$ corresponding to $\mathcal L$ and a choice of global sections $s_0,\ldots,s_n\in\mathcal L(X)$ is a closed immersion if and only if it separates points and tangent vectors, i.e.

1. For closed points $P,Q\in X$ with $P\ne Q$ there exists $s\in V:=\langle s_0,\ldots,s_n\rangle$ with $s(P)=0$ and $s(Q)\ne 0$, using my above notation.
2. For each closed point $P\in X$, the set $\{ s\in V \mid s(P)=0 \}$ spans the $k$-vector space $\mathfrak m_P\mathcal L_P/\mathfrak m_P^2\mathcal L_P$.

The second condition is a bit more subtle, but I found the geometric intuition given at the end of this blog post quite satisfying.