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For a finite generating space $V\subseteq H^0(X,\mathcal{L})$ on a projective scheme, its map to projective space is a closed immersion exactly when, after algebraic closure of the ground field, the displayed evaluation is surjective for every length-two closed subscheme $E$. Two distinct points test point separation; a double point tests a direction in the Zariski tangent space. For the complete section space, this characterizes a very ample line bundle.

A line bundle $\mathcal{L}$ is a globally generated line bundle when the evaluation mapis surjective. Equivalently, its global sections span each fibre of the line bundle; locally at every point, some section is a generator.

Choose a finite generating family $s_0,\dots ,s_n$. On the open set where $s_i$ generates $\mathcal{L}$, the ratios $s_j/s_i$ are regular functions, giving a morphism of schemes into the standard chart $D_{+}(x_i)$ of projective space. The ratios obey the usual transition rules on overlaps, so these chart maps glue toThe tuple is computed using any local trivialization of $\mathcal{L}$; changing that trivialization multiplies all entries by the same invertible function. The construction has ${\phi }^{\ast }\mathcal{O}(1)\simeq \mathcal{L}$ and pulls back the coordinate sections to the chosen $s_i$.

There is a finiteness qualification for an arbitrary $X$: global generation alone need not provide such a finite family. It does if $X$ is quasi-compact, since the open sets on which individual sections generate have a finite subcover. Without that hypothesis, take $X={\coprod }_{m\ge 1}{\mathbb{P}}_k^m$ and let $\mathcal{L}$ restrict to $\mathcal{O}(1)$ on each component. This line bundle is globally generated, but any $N$ global sections have a common zero on a component ${\mathbb{P}}^m$ with $m\ge N$. Thus this is a globally generated line bundle without finite generators. This is finite global generation on a quasi-compact scheme. The finite-family construction is automatic in the projective case asked next.

For projective nonsingular $X$, put $V=⟨s_0,\dots ,s_n⟩$. The length-two criterion for a very ample linear system says that $\phi$ is a closed immersion precisely when, after extending to an algebraic closure, $V$ separates distinct points and tangent directions. Equivalently, the evaluationis surjective for every length-two geometric closed subscheme $E\subseteq X$. Two distinct points give point separation; a nonreduced length-two subscheme supported at one point gives separation of a direction in the Zariski tangent space. For the complete space $V=H^0(X,\mathcal{L})$, this says exactly that $\mathcal{L}$ is a very ample line bundle. For a chosen smaller $V$, it is the chosen linear system of divisors which must be a very ample linear system; mere global generation is insufficient.

Choose a very ample line bundle $A$ on $X$. By the theorem that a high ample twist is very ample, for sufficiently large $m$ bothare very ample. The dual bundle cancels the power of $A$, giving

Write $L\cong H_1\otimes H_2^{\ast }$ as in part (d). Each very ample line bundle is the pullback of $\mathcal{O}(1)$ under a projective embedding. A hyperplane section therefore supplies an effective divisor on a complex manifold $D_i$ with $H_i\cong \mathcal{O}(D_i)$. HenceEvery Picard group class is therefore in the image of the divisor-to-Picard map:

On ${\mathbb{F}}_2$, let $S$ be the negative section of a Hirzebruch surface, so $S^2=-2$, and let $F$ be the fiber class of a Hirzebruch surface, with $F^2=0$ and $S\cdot F=1$. The line bundle $\mathcal{O}(S+3F)$ is very ample. Indeed, the toric ampleness criterion for $aS+bF$ isand on a smooth complete toric variety every ample line bundle is very ample. These inequalities hold for $(a,b)=(1,3)$.

Explicitly, after choosing the standard lattice coordinates for ${\mathbb{F}}_2$, its lattice polytope of a toric divisor isThe monomials indexed by the lattice points of this polygon separate torus orbits and tangent directions, so their Kodaira map is a closed embedding. This directly verifies that the resolved proper toric surface is projective.

A projective embedding realizes a variety as a closed subvariety of projective space. A very ample line bundle gives such an embedding through its global sections.

A very ample divisor is a divisor whose associated line bundle is a very ample line bundle. A divisor is very ample when its complete linear system defines a closed embedding into projective space.