What can be said about a pullback of a very ample line bundle w.r.t birational maps? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:51:29Z http://mathoverflow.net/feeds/question/89725 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89725/what-can-be-said-about-a-pullback-of-a-very-ample-line-bundle-w-r-t-birational-ma What can be said about a pullback of a very ample line bundle w.r.t birational maps? Rami 2012-02-28T03:37:27Z 2012-02-28T03:56:42Z <p>Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$ is birational (but not isomorphism) to its image? Is it ample?</p> http://mathoverflow.net/questions/89725/what-can-be-said-about-a-pullback-of-a-very-ample-line-bundle-w-r-t-birational-ma/89727#89727 Answer by Sándor Kovács for What can be said about a pullback of a very ample line bundle w.r.t birational maps? Sándor Kovács 2012-02-28T03:50:32Z 2012-02-28T03:56:42Z <p>Suppose $\phi$ is a morphism (i.e., defined everywhere) which is birational, but not an embedding. Then there are two cases: </p> <ol> <li>$\phi$ is finite. In this case $\phi^*\mathscr L$ is ample for any ample $\mathscr L$ on the target. An example (pretty much the only one) when this happens is if $\phi$ is the normalization of $\phi(X)$. For instance if $Y$ is any projective singular curve, or for a slightly more interesting example, $Y=Z(xy^2=tz^2)\subset \mathbb P^n$ and $\phi:X\to Y$ is its normalization. </li> <li>$\phi$ is not finite. In this case, (since it's projective) $\phi$ must have positive dimensional fibers, so there exists a curve on which $\phi^*\mathscr L$ is trivial and hence cannot be ample.</li> </ol>