# Space of 0-cycles - definition

I am reading the paper of Farb, Wolfson and Wood "Coincidences between homological densities, predicted by arithmetic" and I have a problem with very first definition (Spaces of 0-cycles): https://arxiv.org/pdf/1611.04563.pdf

Could anybody explain me: 1. How does it work? 2. And why the definition works with the second example, the space of rational based degree $d$ maps from $\mathbb{CP}^1$ to $\mathbb{CP}^{m-1}$?

By definition, an element of $D \in \mathcal{Z}_1^{(d,\ldots,d)}(\mathbb{C})$ is a subset of $md$ (not necessarily distinct) points in $\mathbb{C}$. These points are labelled with colors, and there are exactly $d$ points with each of the $m$ colors. Moreover, there is no point that is labelled every color.
I will be construct the map $f: \mathbb{C}P^1 \to \mathbb{C}P^{m-1}$ associated to $D$. Consider the subsets of $D$ color-by-color. The $d$ points of each color determine the roots of a degree $d$ polynomial $f_i: \mathbb{C} \to \mathbb{C}$ that is unique up to scaling. Let us choose the monic one - the exact choice becomes irrelevant because we will consider these polynomials in projective space and demand that $f(\infty) = [1:1:\cdots:1]$. For concreteness, if $\{z_{i,1},\ldots,z_{i,d}\}$ are the points of a color $i$, we take $f_i(x) = \prod_{j=1}^d (x-z_{i,j})$. The desired map $f$ is then given as $f = [f_1:f_2: \cdots:f_m]$. That is, $$f[x:y] = [\prod_{j=1}^d (x - z_{1,j} y): \prod_{j=1}^d (x - z_{2,j} y): \cdots : \prod_{j=1}^d (x - z_{m,j} y)].$$
One can check that $f(\infty) = f[1:0] = [1:1:\ldots:1]$. Finally, why is this a well-defined regular map? To check this, we need to see that the coordinates don't all simultaneously vanish. But if they all vanished at some $z \in \mathbb{C}$, this would mean that the point $z \in D$ is labelled every color, which is omitted by definition.
• The second condition of the definition is unclear for me: "no point of $X$ is labelled with at least $n$ labels of every color". From this example it means - roughly speaking - that I consider $m$ colours, and there should be no point in $X$ which has 1 (or more) label of every colour? – Igor Sikora Aug 30 '17 at 20:28
• I think you're correct. Another way to state this is to say that an element of $\mathcal{Z}^\vec{d}_n(X)$ consists of $m$ "multi-subsets" of $X$, each of size $d_i$, $1 \leq 1 \leq m$, respectively (counting multiplicity), such that no point of $X$ has multiplicity $\geq n$ in each of the $m$ multi-subsets. – JHF Aug 31 '17 at 14:06