Let $p,d\geq 0$ be two integers and let $X\subseteq\mathbb{P}^N$ be a complex projective variety. Denote the Chow variety of $X$ consisting of $p$-cycles of degree $d$ by $\mathcal{C}_{p,d}(X)$. I'm trying to check that two cycles $\gamma$ and $\gamma^{\prime}$ in $\mathcal{C}_{p,d}(X)$ are algebraically equivalent (as defined at pag. 10 of this) if they lie in the same path connected component of $\mathcal{C}_{p,d}(X)$.
It seems, by taking a look at this paper (proof of Prop. 1.8) that the strategy is to prove that there is a bijective correspondence between morphisms $C\longrightarrow\mathcal{C}_{p,d}(X)$ ($C$ is a smooth curve) and $\mathcal{C}_{p+1,r}(C\times X)$.
I'm looking for a proof of the fact above which avoids any use of scheme theory.
Any help is well accepted.