# Effective versus movable cones of curves

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.

(Q) Is there an example of a smooth projective variety $X$ such that

• $\overline{NE}(X)$ is (finite) polyhedral, but
• $\overline{\mathrm{Mov}}(X)$ is not?

Here are some trivial observation:

## 1

If $X$ is a surface then the two cones are dual to each other and hence they are either both polyhedral or not.

## 2

If $X$ is a Fano variety then

• $\overline{NE}(X)$ is polyhedral by the Cone Theorem, and
• $\overline{\mathrm{Mov}}(X)$ is polyhedral by a result of Barkowski if $\dim \leq4$ (see here) and by [BCHM] in general.
• By the way, Artie Prendergast-Smith has an example here arxiv.org/pdf/0910.5888 of a rational threefold whose nef cone is rational polyhedral but the effective movable cone is not. Would this be an answer to your question? – J.C. Ottem Oct 25 '11 at 8:34
• @JC: yes, this would be an answer. Why don'y you put it as an answer? (In the mean time I realized that this is unfortunately not enough for my initial idea for that other question I started the question with) – Sándor Kovács Oct 25 '11 at 11:34
• p.s: I didn't know about this example, but I was expecting Artie to answer the question. :) – Sándor Kovács Oct 25 '11 at 11:35
• I've added his example in the answer below. I'd also be interested in hearing Artie's thoughts on this, so I hope he joins the discussion. – J.C. Ottem Oct 25 '11 at 12:12
• Another comment: isn't Barkowski's result true in all dimensions because of BDPP (Mov is dual to the pseudoeffective cone) and BCHM (Fanos are Mori dream spaces)? – user5117 Oct 25 '11 at 14:27

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.
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.
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 pseudo-automorphisms (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.
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.