Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.


Does this hold also when $K$ is not algebraically closed?



Nef and pseudo-effective divisors over non algebraically closed fields
I believe the answer is yes. I will use Keeler's paper [Kee03] (and the corrigendum [Kee18]) as a reference for these questions.


First, the theorem of the base holds over arbitrary fields by [Kee03, Theorem 3.6; Kee18, Theorem E2.2] (see also [Cut15, Proposition 2.3]), and hence one can talk about the Néron–Severi space as a finite-dimensional vector space, which Keeler denotes by $V(X)$ in [Kee03, p. 256]. The nef cone $\operatorname{Nef}(X)$ is closed in $V(X)$ [Kee08, p. 256], and we can define the pseudoeffective cone $\overline{\operatorname{Eff}}(X)$ as the closure of the big cone $\operatorname{Big}(X)$ (see, e.g., [Cut15, p. 8]).


Keeler shows that Kleiman's criterion holds over arbitrary fields in [Kee03, Theorem 3.9], and we have
$$\operatorname{Int}\bigl(\operatorname{Nef}(X)\bigr) = \operatorname{Amp}(X).$$
Note that $\operatorname{Nef}(X)$ and $\operatorname{Amp}(X)$ are denoted by $K$ and $K^\circ$ in [Kee03, p. 256].


Now we prove that nef divisors are pseudo-effective. First, by the asymptotic Riemann–Roch theorem [Kol96, Chapter VI, Theorem 2.15], we have
$$\operatorname{Amp}(X) \subseteq \operatorname{Big}(X).$$
Taking closures, we then have
$$\operatorname{Nef}(X) \subseteq \overline{\operatorname{Eff}}(X).$$


References
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[Cut15] Steven Dale Cutkosky, *Teissier's problem on inequalities of nef divisors,* J. Algebra Appl. **14** (2015), no. 9, 1540002, 37 pp. DOI: [10.1142/S0219498815400022](https://doi.org/10.1142/S0219498815400022). MR: [3368254](https://mathscinet.ams.org/mathscinet-getitem?mr=3368254).


[Kee03] Dennis S. Keeler, *Ample filters of invertible sheaves,* J. Algebra **259** (2003), no. 1, 243–283. DOI: [10.1016/S0021-8693(02)00557-4](https://doi.org/10.1016/10.1016/S0021-8693(02)00557-4). MR: [1953719](https://mathscinet.ams.org/mathscinet-getitem?mr=1953719).


[Kee18] Dennis S. Keeler, *Corrigendum to “Ample filters of invertible sheaves”,* J. Algebra **507** (2018), 592–598. DOI: [10.1016/j.jalgebra.2018.03.024](https://doi.org/10.1016/j.jalgebra.2018.03.024). MR: [3807062](https://mathscinet.ams.org/mathscinet-getitem?mr=3807062).


[Kol96] János Kollár, *Rational curves on algebraic varieties,* Ergeb. Math. Grenzgeb. (3), Vol. 32, Springer-Verlag, Berlin, 1996. DOI: [10.1007/978-3-662-03276-3](https://doi.org/10.1007/978-3-662-03276-3). MR: [1440180](https://mathscinet.ams.org/mathscinet-getitem?mr=1440180).


