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 $C\subset Z$ ?



Pushforward of curves
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.**


