Nef and pseudo-effective divisors over non algebraically closed fields

Question

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?

Answer 1

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

[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. 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. 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. 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. MR: 1440180.