Pushforward of curves Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion. 
Let $N_1(X),N_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ respectively, and let $i_{*}:N_1(Z)\rightarrow N_1(X)$ be the pushforward. 
Assume that for a general curve $C\subset Z$ we have that $i_{*}[C] = r[\Gamma]$ where $[\Gamma]$ is a fixed class in $N_1(X)$ and $r\in \mathbb{Q}$.
May we conclude then that $i_{*}[C] = [\Gamma]$ for any curve $C\subset Z$ ?
 A: You can prove this using the "Moving Lemma."  Let $k$ be a field.  Let $Z$ be a projective $k$-scheme that is integral of dimension $d$.  Let $D\subset Z$ be a proper closed subset of $Z$.  Let $C\subseteq Z$ be a closed subscheme that is integral of dimension $d-e$.  Let $W_1,\dots,W_r\subseteq Z$ be integral closed subschemes of dimensions $e_1,\dots,e_r<e$ such that every $W_j$ is disjoint from $C$.
Variant of Moving Lemma. There exists a positive integer $q$, there exist integral closed subschemes $C_1,\dots,C_m\subseteq Z$, and there exist integers $a_1,\dots,a_m$ such that $q[C]$ equals $a_1[C_1]+\dots+a_m[C_m]$ in the Chow group $\text{CH}_{d-e}(Z)$ and such that every $C_i$ is disjoint from every $W_j$ and is not contained in $D$.
Proof.  Choose a closed immersion $j:Z\hookrightarrow \mathbb{P}^n_k$.  Let $\nu:P\to \mathbb{P}^n_k$ be the blowing up of $\mathbb{P}^n_k$ along the closed subscheme $j(C)$.  Denote by $E$ the exceptional divisor of the blowing up, as an effective Cartier divisor.  By the construction of the blowing up as a relative Proj, the associated ideal sheaf $\mathcal{O}_P(-\underline{E})$ is $\nu$-relatively ample.  Thus, for the Cartier divisor class $H$ of $\mathcal{O}_{\mathbb{P}^n}(1)$, for every integer $a\gg 0$, the divisor class $H_{a,1}=a\nu^*H-E$ is ample.  Thus, for every integer $b\gg0$, the divisor class $H_{a,b}=bH_{a,1}$ is very ample.  For the very ample divisor classes $A=H_{2a,b}$ and $B=H_{a,2b}$ with $a\gg 0$ and $b\gg 0$, then $bE$ equals $A-B$, and $2ab\nu^*H$ equals $2A-B$.  
Denote by $Z'\subset P$ the proper transform of $Z$.  Denote by $\phi:Z'\to Z$ the restriction to $Z'$ of $\nu$.  Since $Z$ is integral and $C$ is a proper closed subscheme, also $Z'$ is irreducible and $\phi$ is birational.  As the restriction of a projective morphism, also $\phi$ is projective.  
Denote by $E_Z$ the intersection of $Z'$ with the effective Cartier divisor $E$.  By construction, the support of $E_Z$ is a proper subset of $Z'$.  Thus, $E_Z$ is an effective Cartier divisor in $Z'$.  Every irreducible component of this Cartier divisor has dimension $d-1$.  Similarly, denote by $W'_j$ the inverse image of $W_j$.  Finally, denote by $D'$ the union $\phi^{-1}(D)\cup E_Z$.  
By Bertini's Theorems, there exists an integer $c\gg 0$ and there exist effective Cartier divisors $G_1,\dots,G_{e-1}$ in the linear equivalence class of $cA$ such that the intersection $G_1\cap \dots \cap G_{e-1}\cap E_Z$ is a union of irreducible closed subvarieties $C'_i$ of $E_Z$ each having dimension $d-e$ and such that at least one component dominates $C$.  Thus, the pushforward cycle $\phi_*(G_1\cdots G_{e-1}\cdot E\cdot [Z'])$ is represented by a nonzero, effective cycle of dimension $d-e$ that is supported on $C$.  Therefore, there exists a positive integer $p$ such that the pushforward equals $p[C]$.
On the other hand, using the well-definedness, symmetry, and multilinearity of iterated intersection with divisor classes proved at the beginning of Fulton's Intersection Theory, there is an identity,
$$bG_1\cdots G_{e-1}\cdot E\cdot [Z'] = c^{e-1} A^{e-1}\cdot (A-B)\cdot [Z'] = $$ $$c^{e-1} A^e\cdot [Z'] - c^{e-1} A^{e-1}\cdot B\cdot [Z'].$$  Up to increasing $c$, which also makes $p$ more positive, Bertini's Theorem implies that there exist effective Cartier divisors $G'_1,\dots,G'_e$, resp. $F,$ in the linear equivalence class of $cA$, resp. $cB$, such that the intersections $G'_1\cap \dots \cap G'_e\cap Z'$, resp. $G'_1\cap \dots \cap G'_{e-1}\cap F\cap Z$, are unions of irreducible cycles $C'_i$ of dimension $d-e$ that are disjoint from each $W'_j$ and that are not contained in $D'$.  Thus, increasing $p$ again, the cycle $p[C]$ is rationally equivalent to an integral linear combination of the pushforwards $C_i$ of the cycles $C'_i$.  Each $C_i$ is disjoint from each $W_j$, and it is not contained in $D$. QED.
Remark. This argument makes essential use of the hypothesis that $Z$ is projective. 
Now we can solve the problem.  I will slightly generalize the setup.
Definition. Let $d$ and $e$ be nonnegative integers.  An input datum is a datum $(X,Z,i,D,M)$ of a projective $k$-scheme $X$, an integral, projective $k$-scheme $Z$ of dimension $d$, a $k$-morphism $i:Z\to X$, a proper subvariety $D$ of $Z$, and a $\mathbb{Q}$-subspace $M$ of the $\mathbb{Q}$-vector space of numerical equivalence classes of $(d-e)$-cycles on $X$.  The input datum satisfies the proportionality hypothesis if for every integral closed $(d-e)$-cycle $C\subseteq Z$ with $C\not\subseteq D$, the pushforward $\mathbb{Q}$-numerical equivalence class $i_*[C]$ is in $M$.
Proposition. For every input datum $(X,Z,i,D,M)$ that satisfies the proportionality hypothesis, for every integral closed $(d-e)$-cycle $C\subseteq Z$, the pushforward $\mathbb{Q}$-numerical equivalence class $i_*[C]$ is in $M$.
Proof. By the Moving Lemma, there exists a positive integer $p$ such that $p[C]$ is rationally equivalent to a $\mathbb{Z}$-linear combination of integral, closed $(d-e)$-cycles $C_\alpha$ that are not contained in $D$.  Thus, by the proportionality hypothesis, every $i_*[C_\alpha]$ is in $M$.  Since numerical equivalence factors through rational equivalence, also $pi_*[C]$ is in $M$.  Since $M$ is a $\mathbb{Q}$-vector subspace of the $\mathbb{Q}$-vector space of numerical equivalence classes, also $i_*[C]$ is in $M$. QED.
