Cohomological implications of Twisting by Effective Nef Divisors

Question

It is a standard result due to Serre that for a projective scheme $X$, $D$ an ample divisor on $X$, and $F$ a coherent sheaf, there exists an $N$ such that for all $n\geq N$, one has $H^0(X,F\otimes O_X(nD))\neq 0$ (or better yet, $F\otimes O_X(nD)$ is generated by global sections). In particular, Serre gave a cohomological criterion for ampleness.

I ask a similar question for nef line bundles $O_X(D)$, but with some added conditions to study how far off $D$ is from being ample. I require $D>0$ i.e. it is effective so as to rule out the trivial bundle. Under this hypothesis, is the following property true? Let $D$ be a nef and (numerically?) effective divisor. Then $H^0(X,F\otimes O_X(nD))\neq 0$ for $n\gg 0$ and any coherent sheaf $F$.

I've looked at Lazarsfeld's books and nothing came up. Certainly, there are consequences for cohomology for big and nef divisors such as Kawamata-Viehweg Vanishing and Lazarsfeld's book is riddled with such consequences. These are not what I am searching for. As another example, big divisors themselves have the property that $h^0(X,F\otimes O_X(nD))\geq Cn^{\dim X}$ for $n\gg 0$ and for any coherent sheaf supported on all of $X$ and where $C>0$ is a constant. So my question is a weaker form of this.

Answer 1

I'll give an example of a nef and effective divisor $\tilde{C}$ on a surface $X$ such that $\mathcal{O}_{\tilde{C}} \otimes \mathcal{O}_X(n\tilde{C})$ has no global sections for $n > 0$.

Take $C$ to be an elliptic curve in $\mathbb{P}^2$ and choose $9$ very general points $P_1, \dots, P_9 \in C$ so that $P_1 \oplus \cdots \oplus P_9$ is torsion-free in the group structure $\oplus$ of $C$. Suppose furthermore that the origin $O \in C$ satisfies $\ell|_{C} \sim 3O$ where $\ell$ is the line class in $\mathbb{P}^2$.

Then, consider $\pi: X \to \mathbb{P}^2$ the blowup of those points. Denote $\tilde{C}$ the strict transform of $C$ and $E_i$ the exceptional curves over $P_i$. Then, $\pi$ gives an isomorphism $\pi|_{\tilde{C}}: \tilde{C} \to C$, and as a divisor $\tilde{C} \sim \pi^*C - \sum E_i$. Hence $\tilde{C}^2 = 0$ so it is nef (since it is an effective divisor on a surface).

With the identification of $\pi$ into account we observe $E_i|_{\tilde{C}} = P_i$ and similarly that $$\pi^*C|_{\tilde{C}} = C|_C \sim 3\ell|_C \sim 9O.$$ Hence, by our torsion-free assumption each $\mathcal{O}_{\tilde{C}}(n\tilde{C}) = \mathcal{O}_{\tilde{C}}(\sum n(O - P_i))$ is a nontrivial degree zero line bundle on $\tilde{C}$. In particular, it has no global sections.