Homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$?

For each $m \ge 1$, how do I construct a homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$?

My thoughts so far: I probably want to identify $\mathbb{C}^{m+1}$ with the vector space $\mathbb{C}^m[z, w]$ of degree $m$ homogeneous polynomials in two variables and use that a polynomial in one variable is determined by its roots up to a nonzero constant factor somehow... but from here, I'm kinda of stuck. Could someone help?

EDIT: Progress so far. After reading Eric Wofsey's response, I have the following. Since a projective variety is determined by its homogeneous coordinate ring, it suffices to determine the homogeneous coordinate ring of $(\mathbb{CP}^1)^m/S_m$.

For $(\mathbb{CP}^1)^m$, the coordinate ring is the graded ring $\mathcal{S}$ whose $d$th degree part is the $m$-fold tensor product (over $\mathbb{C}$) of $\mathbb{C}[s, t]_{d}$. That is, $S_d$ consists of polynomials in $s^1, t^1, s^2, t^2, \dots, s^m, t^m$ of multidegree ($d, d, \dots, d$). And $S_m$ acts in the obvious way, i.e. if $\sigma(i) = j$, then $\sigma$ sends $s^i$ to $s^j$ and $t^i$ to $t^j$. The homogeneous coordinate ring of $(\mathbb{CP}^1)^m/S_m$ is just the ring of invariants of the $S_m$ action on $\mathcal{S}$. But I still have two questions.

• Which polynomials are invariant under this action?
• I still don't see how to conclude that Eric Wofsey's map is a homeomorphism. How do I see that Eric Wofsey's map is a homeomorphism?

Solution 1:

You're on the right track; the idea is to take the fact that a polynomial in one variable is determined by its roots and homogenize all the polynomials. You can define a map $f:(\mathbb{CP}^1)^m/S_m\to\mathbb{CP}^m$ by identifying an element $[a,b]\in\mathbb{CP}^1$ with a homogeneous linear polynomial $az+bw$ (defined up to scalar multiples) and then multiplying $m$ such polynomials to get a homogeneous degree $m$ polynomial $c_0z^m+c_1z^{m-1}w+\dots+c_mw^m$ (up to scalar multiples), which can then be identified with $[c_0,\dots,c_m]\in\mathbb{CP}^m$. This is just a homogeneous analogue of the bijection $\mathbb{C}^m/S_m\to \mathbb{C}^m$ which takes $m$ numbers and gives you the coefficients of the monic polynomial which has those numbers as roots. Can you prove that this map is a homeomorphism?