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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 $\mathbb{CP}^{m-1}$?

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I don't know which part of the definition you need clarification, but here is how the second example works. Hopefully this explanation will shed some light on your confusion.

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.

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  • $\begingroup$ 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? $\endgroup$ – Igor Sikora Aug 30 '17 at 20:28
  • $\begingroup$ 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. $\endgroup$ – JHF Aug 31 '17 at 14:06
  • $\begingroup$ Now it's clear - thanks! $\endgroup$ – Igor Sikora Aug 31 '17 at 14:42

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