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 $SL(2,\mathbb C)$?