Nef divisors on surfaces Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:

*

*$h^0(X,hD) = 0$ for $1\leq h\leq h_0-1$;

*$h^0(X,m(h_0D)) = 1$ for all $m \geq 1$;

*$h^0(X,kD) = 0$ if $k$ is not a multiple of $h_0$.

Is there any example of a surface $X$ and a divisor $D$ with this behavior?
Thank you.
 A: Yes, for $h_0=2$, I believe the following construction works.
Let $C$ degree 8 curve in $P^2$ with 16 nodes $p_1,\ldots,p_{16}$ as its singular locus. Let $X$ be the blow-up of $P^2$ along the 16 points, and let $D=4H-E_1-\ldots-E_{16}$. Then $2D$ is represented by an irreducible, smooth curve $\tilde C$ on $X$. Since $D^2=0$, $D$ is nef. If $C$ is generic among 16-nodal curves, one easily checks that the 16 points do not lie on a quartic curve, so $h^0(X,D)=0$. In this generic situation, there should be no relation between the points $p_1,\ldots,p_{16}$ on the curve $\tilde C$ and the hyperplane section $H$, so
$$H^0(\tilde C,\mathcal O_{\tilde C}(mD))=0$$ for all $m\ge 1$. Then by the exact sequence
$$
0\to \mathcal O_{X}((m-2)D)\to \mathcal O_{X}(mD)\to \mathcal O_{\tilde C}(mD)\to 0
$$we get the three properties you want.
A: An example is given by the canonical class $K_S$ of a bielliptic surface $S$, which is torsion of order $h_0=2, 3, 4$ or $6$.
For the definition of a bielliptic surface see for instance Chapter VI of Beauville's book "Complex algebraic surfaces".
