I would like to learn something about the Chow scheme of cycles on an algebraic variety. I am not after an abstract treatment of the moduli problem in full generality, actually I would be happy with a description of it for smooth projective varieties like $\mathbb P^n$. As a start, I would like to know what these Chow schemes look like and how does one define the Hilbert-Chow morphism in this setting - i.e. not from $\textrm{Hilb}\to \textrm{Sym}$ but rather $\textrm{Hilb}\to \textrm{Chow}$.
Of course, if you can provide an answer yourself rather than a reference, you are very welcome! Thanks in advance.