For $N=M=\mathbb{Z}^n$ and $e_0,...,e_n$ standard basis of $\mathbb{Z}^n$ the torus $T=\text{Spec}(K[M])=\text{Spec}(K[x_1^{\pm 1},...,x_n^{\pm 1}])=\mathbb{G}_m^n$, where $x_i=\chi^{e_i}$ with character lattice $M$.
Can someone explain me why I get the torus $T=\text{Spec}(K[M])=\mathbb{G}_m^{n+1}/\mathbb{G}_m$ for the character lattice $M=\{(a_0,....,a_n)\in\mathbb{Z}^{n+1}\mid \sum_{i=0}^na_i=0\}$. The lattice of one-parameter subgroups in this case is $N=\mathbb{Z}^{n+1}/\mathbb{Z}(1,1,...,1)$.
$$\text{Is }\mathbb{G}_m^{n+1}/\mathbb{G}_m=\mathbb{P}^n\backslash \bigcup_{i=0}^nV(x_i)?$$
Thank you in advance.
I try to understand: (Fulton, Introduction to toric varieties)
When you write in Q1 the following: "Is Gnm/Gm=Pn∖⋃ni=0V(xi)?" one could quess there is some type of quotient construction. But dim(Gnm/Gm)=n−1 and the open set Pn−∪V(xi) has dimension n. What "quotient" are you considering in Q1?
Given any abelian group (scheme ) $G$ over $k$, the character group is defined as $Hom_{grp}(G, \mathbb{G}_{m,k})$, where $\mathbb{G}_{m,k}:=Spec(k[t,1/t])$. Which group $G$ are you considering above? You get
$$Hom_{grp}(\mathbb{G}_{a,k}^n,\mathbb{G}_{m,k}) \cong Hom_{k-alg}(k[t,1/t], k[x_1,..,x_n]) \cong k^*.$$
The units in $k[x_i]$ are $k^*$.
It seems there is a well defined map (everything has to be written out in terms of quotients)
$$\rho: \mathbb{G}_{m,k}^{n+1} \rightarrow \mathbb{P}^n- \cup_i V(x_i)$$
defined by
$$\rho(a_0,..,a_n):=[a_0: \cdots : a_n].$$
There is an action $\sigma: \mathbb{G}_{m,k} \times \mathbb{G}_{m,k}^{n+1} \rightarrow \mathbb{G}_{m,k}^{n+1}$ defined by
$$\sigma(u,a):=\sigma(u,(a_0,..,a_n)):=(ua_0,..,ua_n),$$
and $\rho$ "factors": $\rho(\sigma(u,a))=\rho(a)$. In your question you do not specify what you are talking about and what maps you are studying. You should write down all definitions more precisely. Then it will be easier to answer your questions.