Hartshorne Exercise II.7.5.
Establish the following properties of ample and very ample invertible sheaves on a noetherian scheme $X$. $\mathcal L,\mathcal M$ will denote invertible sheaves, and for (d), (e) we assume furthermore that $X$ is of finite type over a noetherian ring $A$.
(a) If $\mathcal L$ is ample and $\mathcal M$ is generated by global sections, then $\mathcal L\otimes\mathcal M$ is ample.
(b) If $\mathcal L$ is ample and $\mathcal M$ is arbitrary, then $\mathcal M\otimes\mathcal L^n$ is ample for sufficiently large $n$.
(c) If $\mathcal L,\mathcal M$ are both ample, so is $\mathcal L\otimes\mathcal M$.
(d) If $\mathcal L$ is very ample and $\mathcal M$ is generated by global sections, then $\mathcal L\otimes\mathcal M$ is very ample.
(e) If $\mathcal L$ is ample, then there is an $n_0>0$ such that $\mathcal L^n$ is very ample for all $n\ge n_0$.
(a) Let $\mathcal F$ be a coherent sheaf on $X$. Let $n_0$ such that $n\ge n_0\implies \mathcal F\otimes \mathcal L^n$ is globally generated. Clearly, the sum of two globally generated sheaves is globally generated because tensor products of global sections correspond to global sections of the tensor product sheaf. It follows immediately that $\mathcal F\otimes (\mathcal L\otimes \mathcal M)^n\cong \mathcal F\otimes \mathcal L^n\otimes \mathcal M^n$ for any $n\ge n_0$.
(b) By part (a) it suffices to show that $\mathcal M\otimes \mathcal L^n$ is globally generated for $n>>0$. This is Hartshorne’s definition of ample.
(c) It suffices to show that some power of $\mathcal L\otimes \mathcal M$ is ample. Then the result follows by (b).
(d) By Serre and (a) we at least know that $\mathcal L\otimes \mathcal M$ is ample, so some power is very ample. We claim this power is 1. Clearly $\mathcal L\otimes\mathcal M$ determines a morphism to $\PP{A}{N}$, given by the product of their linear systems. Since $X$ has a very ample line bundle, it is projective, and therefore it suffices to show that this morphism separates points and tangent vectors. The separation of points is clear (if $s$ separates two points and if $t$ doesn’t vanish then $s\otimes t$ separates those points). For the separation of tangent vectors, if $s_1,…,s_r$ separate tangent vectors at $P$, i.e., their images in $\Omega_{X/k}|_P$ form a basis of $\mathfrak m_P/\mathfrak m_P^2$, then so do the sections $s_1\otimes 1,…,s_r\otimes 1$.
Remark. Although the separation of points and tangent vectors only gives equivalent criteria for a closed immersion over an algebraically closed field, we can see that the only time that this hypothesis is used in the proof of sufficiency is that the residue fields of $\OO_{\PP{k}{N},P}$ and $\OO_{X,P}$ are equal to $k$ for any closed point. Replacing $k$ by $A$ might make it so they are equal to some different field, but they will still always be equal!
(e) Let $m_0$ such that $\mathcal L^{m_0}$ is very ample and $\mathcal L^n$ is globally generated for any $n\ge m_0$. Then by part (d) $\mathcal L^n$ is very ample for all $n\ge 2 m_0=:n_0$.
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