Effective versus movable cones of curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:38:26Z http://mathoverflow.net/feeds/question/79046 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79046/effective-versus-movable-cones-of-curves Effective versus movable cones of curves Sándor Kovács 2011-10-25T03:14:37Z 2011-10-26T21:28:16Z <p>Let $\overline{NE}(X)$ be the closure of the cone generated by the numerical classes of effective curves and $\overline{\mathrm{Mov}}(X)$ the closure of the cone of moving curves. </p> <p>(Q) Is there an example of a smooth projective variety $X$ such that </p> <ul> <li>$\overline{NE}(X)$ is (finite) polyhedral, but </li> <li>$\overline{\mathrm{Mov}}(X)$ is not?</li> </ul> <p>Here are some trivial observation:</p> <h2>1</h2> <p>If $X$ is a surface then the two cones are dual to each other and hence they are either both polyhedral or not.</p> <h2>2</h2> <p>If $X$ is a Fano variety then</p> <ul> <li>$\overline{NE}(X)$ is polyhedral by the Cone Theorem, and </li> <li>$\overline{\mathrm{Mov}}(X)$ is polyhedral by a result of Barkowski if $\dim \leq4$ (see <a href="http://arxiv.org/abs/0902.0206" rel="nofollow">here</a>) and by [BCHM] in general.</li> </ul> http://mathoverflow.net/questions/79046/effective-versus-movable-cones-of-curves/79108#79108 Answer by Artie Prendergast-Smith for Effective versus movable cones of curves Artie Prendergast-Smith 2011-10-25T19:17:19Z 2011-10-25T19:25:57Z <p>As J.C. indicates in the comments, an example for Q1 can be gotten from the variety considered in <a href="http://arxiv.org/abs/0910.5888" rel="nofollow">this paper</a>. This isn't spelled out in the paper, so let me explain it here.</p> <p>First let's change the question into its dual form. The cone of curves is dual to the nef cone, and Boucksom--Demailly--Peternell--Paun showed that the cone of moving curves is dual to the cone of pseudoeffective divisors. So we want to find an example of $X$ such that $Nef(X)$ is rational polyhedral but the cone of pseudoeffective divisors $PsEff(X)$ isn't. </p> <p>I claim the variety $X$ in the linked paper is such an example. Here, $X$ is constructed by blowing up $\mathbf{P}^3$ at the base locus of a general net of quadrics. </p> <p>The variety $X$ is then elliptically fibred over $\mathbf{P^2}$, with the generic fibre having an infinite abelian group (more precisely, rank 8) of sections. Call this group $MW(X)$. Translating by differences of sections gives an action of $MW(X)$ on $X$ by so-called <em>pseudo-automorphisms</em> (meaning birational automorphisms that are isomorphisms in codimension 1). This group action preserves effective divisors, and hence the cone $PsEff(X)$. One can calculate the action fairly explicitly, and in particular one sees the orbit of a divisor $E_i$ (the exceptional divisor of one of the blowups) in N^1(X) is infinite. Now it is easy to see that each $E_i$ spans an extremal ray of $PsEff(X)$, and hence so does any $MW$-translate of $E_i$; since there are infinitely many of these, $PsEff(X)$ has infinitely many extremal rays.</p> <p>On the other hand, it's not hard to show that $Nef(X)$ is rational polyhedral. This is done more or less by brute force: enumerate some curve classes, find the dual cone to the convex hull of those classes (which is then an upper bound for $Nef(X)$), and check that it's spanned by nef classes. Details are in the paper.</p>