Relationship between Line Bundles with isomorphic ring of sections - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:42:28Z http://mathoverflow.net/feeds/question/15407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15407/relationship-between-line-bundles-with-isomorphic-ring-of-sections Relationship between Line Bundles with isomorphic ring of sections Colin Tan 2010-02-16T02:27:00Z 2010-02-17T02:11:28Z <p>Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\Gamma(X_i,mL_i)$. Isomorphism as graded ${\mathbb C}$- algebras.</p> <p>Is there any relationship betweeen $X_1$ and $X_2$? Eg, some morphism between them? How about relationship to $Proj R$?</p> <p>Thanks.</p> http://mathoverflow.net/questions/15407/relationship-between-line-bundles-with-isomorphic-ring-of-sections/15451#15451 Answer by B. Cais for Relationship between Line Bundles with isomorphic ring of sections B. Cais 2010-02-16T14:19:11Z 2010-02-16T14:19:11Z <p>If $X$ is a smooth projective algebraic variety of dimension $d$ over a field and $L$ is an ample line bundle on $X$, then $R=\bigoplus_{m=0}^{\infty} H^0(X,mL)$ is a graded $k$-algebra of dimension $d+1$ and one has $X\simeq \mathrm{Proj}(R)$.</p> http://mathoverflow.net/questions/15407/relationship-between-line-bundles-with-isomorphic-ring-of-sections/15537#15537 Answer by Ben Webster for Relationship between Line Bundles with isomorphic ring of sections Ben Webster 2010-02-17T02:11:28Z 2010-02-17T02:11:28Z <p>To expand on the answer above: as B. Cais says, if the line bundles are ample (which I think follows from positivity by Kodaira), we have a canonical isomorphism $\mathrm{Proj} R_i\cong X_i$. Thus, if the graded rings $R_i$ are isomorphic, then the induced map of Proj's gives an isomorphism $R_1\cong R_2$ carrying one line bundle to the other.</p>