Let $V$ be the standard two-dimensional representation of $SL(2,\mathbb C)$ and let ${\rm Sym}^2V$ be its symmetric square. Let $n$ be a positive integer and consider the following two representations 1) $V^{\oplus n}$ and 2) $({\rm Sym}^2V)^{\oplus n}$ of $SL(2,\mathbb C)$.

**Question 1)** Is there some *explicit* description of the GIT quotient $V^{\oplus n}// SL(2,\mathbb C)$? In particular, is it true that the ring of invariant polynomials is generated by $\frac{n(n-1)}{2}$ quadratic polynomials $vol(v_i,v_j)$ $i\ne j$, where $v=(v_1,\cdots ,v_n)=v\in V^{\oplus n}$ and $vol$ is a volume preserved by $SL(2,\mathbb C)$?

**Question 2).** Is there some *explicit* description of the GIT quotient $({\rm Sym}^2V)^{\oplus n}// SL(2,\mathbb C)$? In particular, is it true that the ring of invariant polynomials is generated by $\frac{n(n+1)}{2}$ quadratic polynomials $(v_i,v_j)$ , where $v=(v_1,\cdots ,v_n)=v\in ({\rm Sym}^2V)^{\oplus n}$ and $(.,.)$ is a symmetric bilinear form on $({\rm Sym}^2V)$ preserved by $SL(2,\mathbb C)$?

so very explicitthat it does not actually use any of the general theory of "Geometric Invariant Theory"! That is important, because Mumford wanted to construct $\mathcal{M}_g$ as a scheme over $\text{Spec}(\mathbb{Z})$, whereas GIT (at that time) only worked over a fixed field. So Mumford uses his explicit quotient construction of $V^{\oplus n}$ over any base as a step in constructing $\mathcal{M}_g$ as a quasi-projective scheme over $\text{Spec}(\mathbb{Z})$. $\endgroup$ – Jason Starr Apr 4 '17 at 15:06