Going furhter on How sections of line bundles rule maps into projective spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:45:50Z http://mathoverflow.net/feeds/question/6355 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6355/going-furhter-on-how-sections-of-line-bundles-rule-maps-into-projective-spaces Going furhter on How sections of line bundles rule maps into projective spaces Csar Lozano Huerta 2009-11-21T06:52:10Z 2010-11-11T14:57:18Z <p>My question is located in trying to follow the argument bellow. </p> <p>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 $d$. </p> <p>-What is the reason for this to happen? -<strong>What is going on with all the sections of $\mathcal{L}^{\otimes d}$ that I am getting as I increase the value of $d$?</strong>. I'll appreciate any comment.</p> <p>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 $d$.</p> <p>Could someone comment further about the word "stabilizes"?.</p> http://mathoverflow.net/questions/6355/going-furhter-on-how-sections-of-line-bundles-rule-maps-into-projective-spaces/6369#6369 Answer by David Lehavi for Going furhter on How sections of line bundles rule maps into projective spaces David Lehavi 2009-11-21T12:47:37Z 2010-11-11T10:34:12Z <ul> <li><p>The sequence stabilizes because any increasing sequence of bounded integers (the dimensions of the images of $X$) stabilizes, but I assume you mean something different.</p></li> <li><p>Suppose that $X\to|L|$ has already stabilized, then the map $X\to|2L|$ decomposes through the map $|L|\to\mathbb{P}\mathrm{Sym}^2 H^0(L)\to\mathbb{P} H^0(2L)$.</p></li> </ul> <p>Where the first map is the Veronese, and the second is a projection.</p> http://mathoverflow.net/questions/6355/going-furhter-on-how-sections-of-line-bundles-rule-maps-into-projective-spaces/45706#45706 Answer by Sándor Kovács for Going furhter on How sections of line bundles rule maps into projective spaces Sándor Kovács 2010-11-11T14:57:18Z 2010-11-11T14:57:18Z <p>Let $R(X,\mathcal L)=\oplus_d H^0(X,\mathcal L^{\otimes d})$ as graded rings, $s_0,\dots s_m\in H^0(X,\mathcal L^{\otimes d})$ and finally $R(X,\mathcal L,s_{\cdot})$ the (graded) subring generated by the $s_0,\dots s_m$ in $R(X,\mathcal L)$. </p> <p>Facts:</p> <p>1) ${s_0,\dots,s_m}$ define a rational map $\sigma: X\dashrightarrow \mathbb P^m$. If ${s_0,\dots,s_m}$ generate $H^0(X,\mathcal L^{\otimes d})$, then $\sigma$ agrees with $\phi_t$. </p> <p>2) If $\vert\{s_0,\dots,s_m\}\vert$ is a basepoint-free linear system, then $\sigma$ is a morphism. and $\sigma(X)\simeq {\rm Proj}\, R(X,\mathcal L, s_{\cdot})$</p> <p>3) If $R(X,\mathcal L,s_{\cdot})$ has the property that for some $a\in\mathbb N$, <code>$R(X,\mathcal L,s_{\cdot})_{ak}=R(X,\mathcal L)_{ak}$</code> for all <code>$k\in \mathbb N, k\gg 0$</code>, then ${\rm Proj}\, R(X,\mathcal L, s_{\cdot})\simeq {\rm Proj}\, R(X,\mathcal L)$ (think of the $a$-uple embedding).</p> <p>4) Since $R(X,\mathcal L)$ is finitely generated, if $s_0,\dots s_m\in H^0(X,\mathcal L^{\otimes d})$ are chosen so that they generate $H^0(X,\mathcal L^{\otimes d})$, then for large enough $d$ the property in 3) will be satisfied, so by 1) and 2) $\phi_d(X)\simeq {\rm Proj}\, R(X,\mathcal L)$ which is independent of $d$, hence it is stabilized.</p>