This concerns a number of basic questions about ample line bundles on a variety $X$
and maps to projective space. I have searched related questions and not found answers, but I apologize if I missed something. I'll work with schemes of finite type over a field $k$ for simplicity.
Background
A quasi-coherent sheaf $F$ on a $k$-scheme $X$ is globally generated if the natural map $H^{0}(X,F)\otimes \mathcal{O}_{X} \rightarrow F$ is a surjection of sheaves. Basically, this says that for any point on $X$, there is at least one section of $F$ that doesn't vanish at that point, so there are enough sections of $F$ to see all the points of $X$. (EDIT: As pointed out in the comments below, this last sentence does not describe a situation equivalent to being globally generated. Perhaps it is better to say that
globally generated means that for each point $x \in X$, $F$ has some rank $r$ at $x$ and globally generated means that there are at least $r$ sections of $F$ that are linearly independent over $x$.)
The notion of globally generated is especially useful when $F=L$ is a line bundle on $X$.
If $V$ is a finite dimensional subspace of $H^{0}(X,L)$ such that $V \otimes \mathcal{O} \rightarrow L$ is surjective, then we get a morphism $\varphi_{V}:X \rightarrow \mathbb{P}(V)$
by the universal property of the projective space $\mathbb{P}(V)$ of hyperplanes in $V$.
Essentially, given a point $x \in X$, we look at the fibre over $x$ of the surjection
$V \otimes \mathcal{O} \rightarrow L$ to get a quotient $V \rightarrow L_{x}$. The kernel
is a hyperplane in $V$, and the morphism $\varphi_{V}$ sends $x$ to that hyperplane as a point in $\mathbb{P}(V)$.
So how to build globally generated sheaves? A line bundle $L$ is called ample
if for every coherent sheaf $F$, $F \otimes L^{\otimes n}$ is globally generated for all large $n$. The smallest $n$ after which this becomes true can depend on $F$.
Finally, a line bundle is called very ample if $L$ is globally generated and $\varphi_{V}$ is an embedding
for some subspace of sections $V$.
There are various properties of and criteria for ample line bundles, which can be found in Hartshorne, for example. What we need for the below questions are the following: $L$ is ample if and only if $L^{m}$ is ample for some $m$ if and only if $L^{n}$
is very ample for some $n$; if $L$ is ample, eventually $L^{k}$ will have sections, be globally generated, be very ample, and have no higher cohomology.
Questions
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Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally generated but not very ample, and of very ample line bundles with higher cohomology?
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Given an ample line bundle $L$, what is the minimal number $k$ so that I can be sure $L^{k}$ has sections, is globally generated, is very ample? Is $k$ related to the dimension of $X$?
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If $L$ is very ample, I can use it to embed $X$ into some projective space. Then by projecting from points off of $X \subset \mathbb{P}^{N}$, I can eventually get a finite morphism $X \rightarrow \mathbb{P}^{d}$, where $d$ is the dimension of $X$. But what if
I just know that $L$ is ample and globally generated? Can I also use it to get such a finite morphism to $\mathbb{P}^{d}$?
Best Answer
1. Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally generated but not very ample, and of very ample line bundles with higher cohomology?
On a curve of genus $g$, a general divisor of degree $d \le g-1$ has no sections. Of course, if $d>0$ then it is ample.
$K_X$ on a hyperelliptic curve is globally generated but not very ample.
Look at $L=\mathcal O(1)$ on a plane curve of genus $d$. Then from $$ 0\to \mathcal O_{\mathbb P^2}(1-d) \to \mathcal O_{\mathbb P^2}(1) \to \mathcal O_C(1)\to 0$$
you see that $H^1(\mathcal O_C(1))=H^2(\mathcal O_{\mathbb P^2}(1-d))$ which is dual to $H^0(\mathcal O_{\mathbb P^2}(d-4))$. So that's nonzero for $\ge4$.
2. Given an ample line bundle $L$, what is the minimal number $k$ so that I can be sure $L^k$ has sections, is globally generated, is very ample? Is $k$ related to the dimension of $X$?
Again, just look at the divisor of a degree 1 on a curve of genus $g$. You need $k\ge g$, so you see that there is no bound in terms of the dimension.
It turns out that a better right question to ask is about the adjoint line bundles $\omega_X\otimes L^k$ ($K_X+kL$ written additively). Then the basic guiding conjecture is by Fujita, and which says that for $k\ge \dim X+1$ the sheaf is globally generated, and for $k\ge \dim X+2$ it is very ample. This is proved for $\dim X=2$, proved with slightly worse bounds for $\dim X=3$. For higher dimensions the best result is due to Angehrn-Siu who gave a quadratic bound on $k$ instead of linear. There are some small improvements for some special cases.
3. If $L$ is very ample, I can use it to embed $X$ into some projective space. Then by projecting from points off of $X\subset \mathbb P^N$, I can eventually get a finite morphism $X\to \mathbb{P}^d$, where $d$ is the dimension of $X$. But what if I just know that $L$ is ample and globally generated? Can I also use it to get such a finite morphism to $\mathbb P^d$?
But of course $L$ gives a morphism $f$, and it follows that $f$ is finite: $f$ contacts no curve so $f$ is quasifinite, and $f$ is projective (since $X$ was assumed to be projective). And quasifinite + proper = finite.