TITLE: Ample line bundles, sections, morphisms to projective space
QUESTION [19 upvotes]: 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

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?
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 $X$?
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 $\mathbb{P}^{d}$?

REPLY [11 votes]: The answers already given seem to settle your questions, but I'd like to emphasise one point that hasn't been mentioned.
Suppose X is a projective variety, and $\pi: X' \rightarrow X$ any morphism which contracts at least one curve in X'. (For example, it could be the blowup of any subvariety of X of codimension at least 2.) Then for any very ample line bundle L on X, the pullback $\pi^*L$ is globally generated but not ample, because it has degree zero on any curve contracted by $\pi$. (So to get a simple example for the first part of your first question, let $\pi$ be the blowup of P^2 in a point, and L the class of a line.)
The point I want to make is that this is not just a class of examples, but in fact the general case. To see this, suppose X is a projective variety and L a line bundle which is globally generated but not ample. Then L defines a morphism $f_L: X \rightarrow Y \subseteq \mathbb P^n$ to some subvariety of projective space with $L = f_L^* H$, the pullback of the hyperplane class (which is very ample on Y). Now since L is not ample, the morphism $f_L$ must contract some curve on X: in other words, the Stein factorisation of $f_L$ consists of a nontrivial contraction followed by a finite morphism. (Here a contraction means a projective morphism with connected fibres.) 
The moral is that for X any projective variety, globally generated but not ample line bundles on X are the "same thing" as nontrivial contraction morphisms $\pi: X \rightarrow Y$. (The correspondence can be made more precise by considering the face structure of the nef cone of X, but that's not really relevant to your question.)
Finally, a word on terminology: a line bundle L on X of which some power $L^k$ is globally generated is called (by some people at least) semi-ample. For some purposes this turns out to be a better notion than globally generated, just as ample is a better notion than very ample.