Let $D$ be a pseudo-effective $\mathbb{R}$-divisor on a rational surface. Can we find an example that the numerical class of $D$ contains no effective divisor?
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2$\begingroup$ "Numerically effective class" and "numerical class of effective divisor" are not the same things. It is better to change the title of your question in accordance with the question itself. $\endgroup$– SashaCommented Oct 23 at 6:08
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1$\begingroup$ I think you are asking whether the Mori cone NE(X) on a surface X is closed. A ruled surface counterexample due to Mumford is given in Lazarsfeld's Positivity in Algebraic Geometry I, Example 1.5.1. $\endgroup$– Evgeny ShinderCommented Oct 23 at 6:30
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2$\begingroup$ @EvgenyShinder I think the question is a bit different: is the closed $\mathbb{R}_{\geq 0}$-cone of pseudoeffective divisors (i.e., the closure of the big cone) spanned by effective divisors? $\endgroup$– Jason StarrCommented Oct 23 at 11:20
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2$\begingroup$ @R.vanDobbendeBruyn Mumford's example is a P1-bundle over a curve; producing non-effective divisor rely producing a certain stable certain rank 2 bundle on C. Anyway, P1-bundles which are rational must be Hirzebruch surfaces, hence toric, so all cones are rational polyhedral. $\endgroup$– Ennio Mori coneCommented Oct 23 at 18:44
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Nagata's famous counterexample to Hilbert's 14th problem seems to be what you are after. In this example, Nagata considers the blow-up $X$ of $P^2$ at 16 very general points $p_1,\ldots,p_{16}$. So $X$ is a smooth rational surface with Picard group $\mathbb{Z}^{17}$. If $H$ denotes the pullback of the hyperplane class in $P^2$, and $E_1,\ldots,E_{16}$ denote the exceptional divisors, then the class $$ D=4H-E_1-\ldots-E_{16} $$is nef, hence pseudoeffective. Indeed, when the 16 points are a complete intersection of two degree 4 curves $V(F)$ and $V(G)$, it is easy to see that $D\cdot C\ge 0$ for every curve $C$, by considering the linear system of curves $\{sF+tG=0\}$ for varying $s$ and $t$. Therefore, by a specialization argument, the class $D$ remains nef on the very general blow-up as well.
However, Nagata showed, using a similar specialization argument, that no multiple of $D$ is effective.
I believe this is explained in the paper
M. Nagata. On the fourteenth problem of Hilbert. Proc. ICM Edinburgh (1958), 459–462.
