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My question is located in trying to follow the argument bellow.

Given a normal algebraic variety $X$, and a line bundle $\mathcal{L}\rightarrow X$ which is ample, then eventually such a line bundle will have enough section to define an embedding $\phi:X\rightarrow \mathbb(H^0(X,\mathcal{L}^{\otimes d}))=\mathbb{P}^N$ (notice that $N$ depends on $d$). However, if the line bundle is NOT ample we can still say something about the existence of a certain map $\phi_d$; the so-called Iitaka fibration. I'll omit some details, but the argument of the construction goes (more or less) as follows. Suppose in the first place that $\mathcal{L}$ is base-point base. That is to say, that there are no points (or a set) of $X$ which all the hyperplanes of $\mathbb{P}^N$ pass through. Then such a line bundle will define a linear system $|\mathcal{L}^d|$ which gives rise to a morphism $\phi_d:X\rightarrow \phi_d(X)\subset\mathbb{P}^N$ (again $N$ depends on $d$). Such a map may not be an embedding, however $\phi_d:X\rightarrow \phi(X)$ is an algebraic fiber space. My question is the following. As I increase the value $d$ the image $\phi_d(X)\subset \mathbb{P}^N$ may change, however, as a matter of fact such an image "stabilizes" as $d$ gets larger. Meaning that if $d$ is large enough, the image if $\phi_d$ is the "same" regardless $d$.

-What is the reason for this to happen? -What is going on with all the sections of $\mathcal{L}^{\otimes d}$ that I am getting as I increase the value of $d$?. I'll appreciate any comment.

As a result, due to the fact that after a while we no longer care about the value of $d$, we can associate the space $X\rightarrow \phi(X)$ to the line bundle $\mathcal{L}$. Here, the variety $\phi(X)$ no longer depends on $d$.

Could someone comment further about the word "stabilizes"?.

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# Going furhter on How sections of line bundles rule maps into projective spaces

My question is located in trying to follow the argument bellow.

Given a normal algebraic variety $X$, and a line bundle $\mathcal{L}\rightarrow X$ which is ample, then eventually such a line bundle will have enough section to define an embedding $\phi:X\rightarrow \mathbb(H^0(X,\mathcal{L}^{\otimes d}))=\mathbb{P}^N$ (notice that $N$ depends on $d$). However, if the line bundle is NOT ample we can still say something about the existence of a certain map $\phi_d$; the so-called Iitaka fibration. I'll omit some details, but the argument of the construction goes (more or less) as follows. Suppose in the first place that $\mathcal{L}$ is base-point base. That is to say, that there are no points or a set which all the hyperplanes of $\mathbb{P}^N$ pass through. Then such a line bundle will define a linear system $|\mathcal{L}^d|$ which gives rise to a morphism $\phi_d:X\rightarrow \phi_d(X)\subset\mathbb{P}^N$ (again $N$ depends on $d$). Such a map may not be an embedding, however $\phi_d:X\rightarrow \phi(X)$ is an algebraic fiber space. My question is the following. As I increase the value $d$ the image $\phi_d(X)\subset \mathbb{P}^N$ may change, however, as a matter of fact such an image "stabilizes" as $d$ gets larger. Meaning that if $d$ is large enough, the image if $\phi_d$ is the "same" regardless $d$.

-What is the reason for this to happen? -What is going on with all the sections of $\mathcal{L}^{\otimes d}$ that I am getting as I increase the value of $d$?. I'll appreciate any comment.

As a result, due to the fact that after a while we no longer care about the value of $d$, we can associate the space $X\rightarrow \phi(X)$ to the line bundle $\mathcal{L}$. Here, the variety $\phi(X)$ no longer depends on $d$.

Could someone comment further about the word "stabilizes"?.