Take a minimal surface $S$ of general type with $p_g=1$, $q=0$ and zero torsion.

Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.

In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group coincides with the Picard group and numerically equivalent divisor are linearly equivalent.

Examples of these surfaces, with $K^2=2$, are given in the paper of Debarre and Catanese

"Surfaces with $K^2=2$, $p_g=1$, $q=0$",

J. reine angew. Math. 395 (1989), 1-55.

Take a minimal surface $S$ of general type with $p_g=1$, $q=0$ and zero torsion.

Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.

In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group $\textrm{NS}(S)$ coincides with the Picard group $\textrm{Pic}(S)$ and any two numerically equivalent divisors on $S$ are linearly equivalent.

Examples of these surfaces, with $K^2=2$, are given in the paper of Debarre and Catanese

"Surfaces with $K^2=2$, $p_g=1$, $q=0$",

J. reine angew. Math. 395 (1989), 1-55.

Take a minimal surface $S$ of general type with $p_g=1$, $q=0$ and zero torsion.

Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.

In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group coincides with the Picard group and numerically equivalent divisor are linearly equivalent.

Examples of these surfaces, with $K^2=2$, are given in the paper of Debarre and Catanese

"Surfaces with $K^2=2$, $p_g=1$, $q=0$",

J. reine angew. Math. 395 (1989), 1-55.

Take a minimal surface $S$ of general type with $p_g=1$, $q=0$ and zero torsion.

Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.

In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group coincides with the Picard group and numerically equivalent divisor are linearly equivalent.

Examples of these surfaces, with $K^2=2$, are given in the paper of Debarre and Catanese

"Surfaces with $K^2=2$, $p_g=1$, $q=0$",

J. reine angew. Math. 395 (1989), 1-55.