$r$-fold product of projective space

Question

I try to understand the $r$-fold symmetric product of $\mathbb{P}^n_k$. It is defined as the fibre product (in the category of schemes) $\mathbb{P}^n_k \times _k \dots \times _k \mathbb{P}^n_k$ ($r$ times) quotient by the action of the symmetric group $S_r$ which permutes coordinates. But what are coordinates here? How can we identify a tuple with $r$ entries from $\mathbb{P}^n_k$ as an element in $\mathbb{P}^n_k \times _k \dots \times _k \mathbb{P}^n_k$? Gives this just a closed point or something like that?

Thank you.

Answer 1

I'm not sure about schemes specifically, because I don't know if the fibered product is actually a categorical fibered product, but in a general category $C$ with finite products, you always have a permutation representation $S_r \to Aut(X^r)$.

It is given in the following way (up to a $^{-1}$ that may be missing to give the right variance to the action) : if $X$ is your object and $\sigma$ your permutation, then you have projection maps $\pi_{\sigma(i)}: X^r\to X$ for each $i$, that assemble as a map $X^r\to X^r$ by the universal property of a product. It is then easy to check (using the uniqueness in the universal property) that this gives an action (perhaps a right action instead of a left one)

Now fibered product over a fixed object $S$ are just products in the comma category $C/S$, so what I said still applies in this comma category.

Answer 2

First, recall $\newcommand{\Proj}{\operatorname{Proj}}\mathbb{P}^n_k=\Proj k[x_0,\dots,x_n]$. The product of $r$ copies of $\mathbb{P}^n_k$ over $\operatorname{Spec} k$ is $\Proj (k[x_0,\dots,x_n]^{\otimes r}/I)$, where $I$ is the Segre embedding ideal $(x_i^{(a)}x_j^{(b)}-x_j^{(a)}y_i^{(b)}:0\leq i\leq j\leq n, 1\leq a\leq b\leq r)$. Then the categorical (or GIT) quotient $(\mathbb{P}^n_k)^r // S_r$ is the Proj of the invariants $\Proj((k[x_0,\dots,x_n]^{\otimes r}/I)^{S_r})$. We have $$ k[x_0,x_1,\dots,x_n]^{\otimes r}=k[\{x_{i_1}^{(1)}x_{i_2}^{(2)}\dots x_{i_r}^{(r)}\mid 0\leq i_j\leq n\text{ for all }j\}] $$ and $S_r$ acts on the generator by $$ \sigma\cdot x_{i_1}^{(1)}x_{i_2}^{(2)}\dots x_{i_r}^{(r)} = x_{i_1}^{(\sigma1)}x_{i_2}^{(\sigma2)}\dots x_{i_r}^{(\sigma r)} $$ If $n=1$, it is easy to see the ring of invariants is analogous to elementary symmetric polynomials $$ k\left[\left\{\sum_{J\in\{1,\dots,r\}^{(m)}}\prod_{j\in J}x_1^{(j)}\prod_{j'\notin J}x_0^{(j')}\right\}_{m=0,1,\dots,r}\right] $$ so we see $\operatorname{Sym}^r\mathbb{P}^1_k=\mathbb{P}^r_k$. For $n>1$, we can similarly write down the generators, but we don't have as easy an identification.