Why the GIT quotient $\mathbb{A}^m//\mathbb G_{m}$ is empty? Let $\mathbb{A}^m ={\text{Spec }}\mathbb{C}[x_1,\dots,x_m]$, and the multiplicative group $\mathbb G_m \cong \mathbb{C}^*$ acts on $\mathbb{C}[x_1,\dots,x_m]$ by
$$\lambda_t (x_1,\dots,x_m)= (t^{a_1}x_1,\dots,t^{a_m}x_m)$$
suppose for all $i, a_i >0$.
Then it is claimed the GIT quotient $\mathbb{A}^m//\mathbb G_{m}$ is empty, but I did not see why this is true. I am not good at GIT stuff, and I don't know why this is different from
$$\mathbb{A}^m//G_{m} = \text{Spec }\mathbb{C}[x_1,\dots,x_m]^{G_m} = \text{Spec }\mathbb{C}?$$
 A: Denote $A=\mathbb C[x_1, \ldots, x_m]$, $X=\operatorname{Spec} A$ and $G=\mathbb G_m$. First, there is an affine quotient $X/G := \operatorname{Spec} \left( A^G \right)$, which in this particular case is a point.
Second, there is a GIT quotient
$$X //_{\mathcal L} G=\operatorname{Proj} \left( \bigoplus_{k \geqslant 0} H^0(X, \mathcal L^{\otimes k})^G \right),$$
which depends on the choice of $G$-linearized line bundle $\mathcal L$ on $X$. There are no non-trivial line bundles on $X=\mathbb A^m$, and $G=\mathbb G_m$ is connected, therefore $\mathcal L=\mathcal O(\chi)$ for a character $\chi: G \to \mathbb G_m$. It implies that
$$H^0 \left( X, \mathcal O(\chi^k) \right)^G=A^G_{\chi^k}:=\{ f \in A: f(gx)=\chi^k(g) f(x) \},$$
a subspace of $\chi^k$-semi-invariant functions.


*

*If $\chi(g)=1$ for all $g \in G$, then
$$X //_{\mathcal O(\chi)} G=\operatorname{Proj} \left( A^G[t] \right)=\operatorname{Spec} \left( A^G \right)$$
is an affine quotient again. 

*Instead one should take $\chi(g)=g$, then
$$X //_{\mathcal O(\chi)} G=\operatorname{Proj} A$$
with the usual grading on $A$, that is a weighted projective space.

*More generally, if $\chi(g)=g^k$ with $k>0$, then the grading is $\deg x_i=k  \cdot a_i$, but the quotient is still a weighted projective space.

*At last, if $\chi(g)=g^k$ with $k<0$, then
$$X //_{\mathcal O(\chi)} G=\operatorname{Proj} \left( A^G \right)$$
is indeed empty.
For details, see Mukai An introduction to invariants and moduli, chapter 6.1.
