How can I compute the presentation of a finite group quotient of a quasiprojective variety?

Question

I want to learn how to compute examples of quotient varieties using finite group quotients and quasi-projective varieties. Are there any tools which make computing rings of invariants "by hand" easy? For example, I want to compute quotients of the K3 $$ \frac{\mathbb{C}[x_0,x_1,x_2,x_3]}{(x_0^4 + x_1^4 + x_2^4 + x_3^4)} $$ using various compositions of the group action \begin{align*} x_0 \to \zeta_4x_0 && x_1 \to \zeta_4x_1 &&x_2 \to \zeta_4x_2 && x_3 \to \zeta_4x_3 \end{align*}

Answer 1

This is a particularly easy case. The invariants are generated by monomials and a monomial $x_0^ax_1^bx_2^cx_3^d$ is invariant if and only if $a+b+c+d$ is a multiple of 4. The rest should be clear.