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Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the disjoint union of all the closed points $x\in X$ of degree $[\kappa(x):k]<N$ in $X$. For $n< N$, we have a bijection $${\rm Sym}^ni: {\rm Sym}^nZ(k)\xrightarrow{\cong}{\rm Sym}^nX(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then ${\rm Sym}^nX(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in Hilbert and Chow Schemes of Points, Symmetric Products and Divided Powers.

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the (finite) disjoint union of ${\rm Spec}(\kappa(x))$ for all the closed points $x\in X$ of degree $[\kappa(x):k]<N$. For $n< N$, we have a bijection $${\rm Sym}^ni: ({\rm Sym}^nZ)(k)\xrightarrow{\cong}({\rm Sym}^nX)(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then $({\rm Sym}^nX)(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in Hilbert and Chow Schemes of Points, Symmetric Products and Divided Powers.

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the disjoint union of all the closed points $x\in X$ of degree $[\kappa(x):k]<N$ in $X$. For $n< N$, we have a bijection $${\rm Sym}^ni: {\rm Sym}^nZ(k)\xrightarrow{\cong}{\rm Sym}^nX(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then ${\rm Sym}^nX(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in the paper here.

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the disjoint union of all the closed points $x\in X$ of degree $[\kappa(x):k]<N$ in $X$. For $n< N$, we have a bijection $${\rm Sym}^ni: {\rm Sym}^nZ(k)\xrightarrow{\cong}{\rm Sym}^nX(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then ${\rm Sym}^nX(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in Hilbert and Chow Schemes of Points, Symmetric Products and Divided Powers.

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the disjoint union of all the closed points $x\in X$ of degree $[\kappa(x):k]<N$ in $X$. For $n< N$, we have a bijection $${\rm Sym}^ni: {\rm Sym}^nZ(k)\xrightarrow{\cong}{\rm Sym}^nX(k).$$
2. If $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then ${\rm Sym}^nX(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in the paper here.

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):

1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the disjoint union of all the closed points $x\in X$ of degree $[\kappa(x):k]<N$ in $X$. For $n< N$, we have a bijection $${\rm Sym}^ni: {\rm Sym}^nZ(k)\xrightarrow{\cong}{\rm Sym}^nX(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then ${\rm Sym}^nX(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.

I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!

For the construction of symmetric products of varieties, one can see Section 3.1 in the paper here.