Cone of effective divisors! - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:38:35Z http://mathoverflow.net/feeds/question/67959 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67959/cone-of-effective-divisors Cone of effective divisors! Mohammad F.Tehrani 2011-06-16T15:37:33Z 2011-06-16T18:53:55Z <p>Let $X$ be a smooth simply connected projective variety of dimension $n$ (over complex numbers of course). For such $X$ we have two famous cones which are cone of effective curves and ample cone and are dual to each other.</p> <p>Question: Is there any thing as Cone of effective divisors? Is there any problem to define such a thing? Has any body studied that? For surfaces, it is just cone of effective curves. So the smallest dimension at which we would get some thing new is three.</p> http://mathoverflow.net/questions/67959/cone-of-effective-divisors/67987#67987 Answer by Dave Anderson for Cone of effective divisors! Dave Anderson 2011-06-16T18:53:55Z 2011-06-16T18:53:55Z <p>As mentioned in the comments, the <em>(pseudo)effective cone</em> $\overline{\mathrm{Eff}}(X)$, defined as the closure of the cone of all effective divisors on $X$, is certainly an object of study, and Lazarsfeld's book is a good reference. Your complaint that he doesn't say much about its structure is surely related to the fact that so little is known! Here are a few general things I'm aware of:</p> <ul> <li><p>The interior of the effective cone is the <em>big cone</em>, i.e., the cone of line bundles with positive volume.</p></li> <li><p>The dual of the effective cone is the <em>cone of moveable curves</em>, see Boucksom-Demailly-Paun-Peternell.</p></li> <li><p>As part of their work on the minimal model program, Birkar-Cascini-Hacon-McKernan prove that log Fano varieties have finitely generated effective cones.</p></li> </ul> <p>And here are a couple specific instances where one knows more:</p> <ul> <li><p>When $X$ admits an action by a solvable group with a dense orbit, the effective cone is generated by the components of the complement of the orbit. (This works when $X$ is, e.g., a toric variety or a Schubert variety.)</p></li> <li><p>There's been a lot of recent work on the case $X=\overline{M}_{0,n}$, see e.g., Hu-Keel, Hassett-Tschinkel, Castravet-Tevelev.</p></li> </ul>