The property of line bundle corresponding to birational map For a variety, if it has a base point free line bundle, then one can define a morphism from the variety to a $\mathbb{P}^n$. And if a variety is a projective variety(in the sense of Hartshorne), it is equivalent to have a very ample line bundle.
In the same vein, I was wondering if there are properties of line bundle corresponding to birational map. To be precisely:
Let $V$ be a variety, $L$ be a line bundle on $V$. Which conditions are needed to ensure the $map$ defined by $L$ (i.e. using all the basis of $H^0(V,L)$)is a birational map to its image in $\mathbb{P}^n$? Here $n=dim\ H^0(V,L)-1$.
 A: I don't have a satisfying answer. Anyway here is one. 
Let $L$ be a basepoint-free line bundle on a projective variety $V$ over a field $k$. Let 
$$\pi : V \to \mathbb P(H^0(V, L))\simeq \mathbb P^n$$ 
be the morphism associated to the complete linear system $|L|$. If we choose a basis $s_0, \dots, s_n$ of $H^0(V, L)$, then set-theoretically 
$$\pi(v)=[s_0(v):\dots : s_n(v)].$$ 
If $F(X_0,\dots, X_n)$ is a homogeneous polynomial of degree $m$, then $\pi^{-1}(D_+(F))$ is $V_f$ where $f=F(s_0,\dots, s_n)\in H^0(V, L)^{\otimes m}$ and 
$$ V_f:=\{ v\in V \mid f(v)\ne 0\}.$$
The morphism $V_f\to D_+(F)$ is given by 
$$ k[T_0, \dots, T_n]_{(F)} \to O_V(V_f), \quad P/F^r \mapsto P(s)/F(s)^r.$$ 
We know that if $L$ is very ample, then for any section $f\in H^0(V, L)$, $V_f$ is an affine open subset and the above homomorphism is surjective. The same is true if we take $f$ in $H^0(V, L)^{\otimes m}$ for any $m\ge 1$.  
In general $\pi$ is birational if and only if there exists a dense open subset $U$ of $\mathbb P^n$ such that $\pi: \pi^{-1}(U) \to U$ is a closed immersion (an isomorphism from $\pi^{-1}(U)$ onto $U\cap \pi(X)$). The complement of $U$ is contained in a hypersurface $V_+(F)$ of deree $m$. Shrinking $U$ if necessary, we can suppose $U=D_+(F)$. Then $\pi^{-1}(U)=V_f$ where $f=F(s)$ as above. The condition for $V_f\to U$ to be a closed immersion is: $V_f$ is affine and $O_V(V_f)$ is generated, as $k$-algebra, by $H(s)/f(s)$ where $H$ runs through monomials of degree $m$. 
Summarizing the above discussion, we can say that $\pi$ induces a birational morphism from $V$ to its image if and only if there exists $m\ge 1$ and $f\in H^0(V, L)^{\otimes m}$ such that:


*

*$V_f$ is affine,

*$O_V(V_f)$ is generated by $H(s_0,\dots, s_n)/f$ where $H$ are the monomials of degree $m$. 


Everything remains true if we replace $H^0(V, L)$ with a vector subspace generating $L$. 
