Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:13:09Z http://mathoverflow.net/feeds/question/7914 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7914/torsion-line-bundles-with-non-vanishing-cohomology-on-smooth-acm-surfaces Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces Hailong Dao 2009-12-05T22:39:11Z 2009-12-09T18:13:02Z <p>I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$ (such thing is called an ACM surface, I think) and a <S>globally generated</S> line bundle $L$ such that $L$ is torsion in $Pic(X)$ and $H^1(L) \neq 0$. </p> <p>Does such surface exist? How can I construct one if it does exist? What if one ask for even nicer surface, such as arithmetically Gorenstein? If not, then I am willing to drop smooth or globally generated, but would like to keep the torsion condition. </p> <p>More motivations(thanks Andrew): Such a line bundle would give a cyclic cover of $X$ which is not ACM, which would be of interest to me. I suppose one can think of this as a special counter example to a weaker (CM) version of purity of branch locus. </p> <p>To the best of my knowledge this is not a homework question (: But I do not know much geometry, so may be some one can tell me where to find an answer. Thanks.</p> <p>EDIT: Removed the global generation condition, by Dmitri's answer. I realized I did not really need it that much. </p> http://mathoverflow.net/questions/7914/torsion-line-bundles-with-non-vanishing-cohomology-on-smooth-acm-surfaces/8355#8355 Answer by Dmitri for Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces Dmitri 2009-12-09T14:51:01Z 2009-12-09T14:51:01Z <p>Let us show that a globaly generated torsion line bundle $L$ on a (compact) complex surface is trivial. Ideed, a globally generated line bundle has at least one section, say $s$. Let us take it. If $s$ has no zeros, then $L$ is trivial. But if $s$ vanishes somewhere then any positive power $L^n$ has a section $s^n$ that vanishes at the same points. So any power of $L$ is not trivial, i.e. $L$ is not a torsion bundle, contradiction.</p> <p>Notice that we did not use the fact that the surface is smooth. And we also did not use the fact that we work with a surface...</p>