Let $X$ be a projective variety over a field $k$, and $\dim X \geq 2$. By a curve $C$ on $X$, I mean a proper, reduced subscheme of $X$ of dimension $1$.


(1) If $C$ is an irreducible curve on $X$, then is $C$ numerically equivalent to some $\sum n\_i C\_i$ with $n\_i > 0$, and $C\_i$ smooth curves?


(2) If $D$ is a Cartier divisor on $X$, and $D\cdot C \geq 0$ for any smooth curve $C$ on $X$, then is $D$ necessarily to be a nef divisor?


Certainly, if (1) holds, then (2) holds.


(I ask this question because I am curious why people usually call a nonconstant morphism $C \to X$ with $C$ being a smooth curve to be a curve on $X$, rather than a smooth curve $C$ exactly sitting inside $X$.)



Curves on varieties, and a criterion for nef divisor
The answer to question $(1)$ is **no**. In fact, take an irreducible, nodal cubic curve $A \subset \mathbb{P}^2$ and take $10$ points $p\_1, \ldots, p\_{10}$ on it, different from the node. Let $X$ be the blow-up of $\mathbb{P}^2$ at the points $p\_i$ and $C$ the strict transform of $A$ in $X$. Then $C$ is an irreducible nodal curve isomorphic to $A$ and such that $C^2=-1$. 


This implies that $C$ is *isolated* in its numerical equivalence class. Indeed, assume that $D$ is effective and numerically equivalent to $C$; since $CD =-1$ and $C$ is irreducible, it follows that $C$ is a component of $D$. Then $D=C+Z$, where $Z$ is effective and numerically trivial on $X$; so $Z=0$ and $C=D$. In particular, no positive linear combination of smooth curves can be numerically equivalent to $C$. 


This also shows that the answer to question $(2)$ is **no**. In fact, the curve $C$ in the example above clearly has non-negative intersection with any irreducible curve on $X$ different from it, in particular it has non-negative intersection with all the smooth curves. However, $C$ is *not* nef since $C^2 =-1 <0$. 


