Let $X$ be a projective variety over $k$, 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 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 $X$.)

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 $X$.)

Let $X$ be a projective variety over $k$, 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 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 $X$.)

Let $X$ be a projective variety over $k$, 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 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 $X$.)