In his Paper [Sc], Scott proves that the coordinate ring of the grassmannian $ Gr(k,n) \subset \mathbb{P}\Lambda^k\mathbb{C}^n$, or rather of its affine cone, is a cluster algebra of geometric type. In section 6 (page 33) of his paper he shows that each cluster yields a toric chart, more precisely:
Let us denote the affine cone by $X=X(k,n) \subset \Lambda^k\mathbb{C}^n$ and define $N := (k-1)(n-k-1)$.
Now fix a cluster $ x = \{x_1, ... , x_N\} \subset \mathbb{C}[X]$ and let $c = \{ \Delta_{K_1}, ... , \Delta_{K_n} \}$ be the set of Plücker coordinates corresponding to the "intervals" $ K_i = [i,i+k] \subset \{1,...,n\}$ (taken modulo $k$).
If $U \subset X$ is the open subvariety obtained by omitting the zero loci of the cluster variables $x_1 , ... , x_N$ and the Plücker coordinates $\Delta_{K_1}, ... , \Delta_{K_n} $ then the map
$$ \Phi_x : U \to (\mathbb{C}^*)^{N+n}; p \mapsto (x_1(p), ... , x_N(p), \Delta_{K_1}(p), ... , \Delta_{K_n}(p)) $$ is biregular.
My question is about his proof that this map is surjective:
Every Plücker coordinate can be uniquely expressed as a Laurent polynomial in the cluster variables $x \cup c $, so given a point $a \in (\mathbb{C}^*)^{N+n} $, one can define the vector $v_a \in \Lambda^k\mathbb{C}^n$ whose Plücker coordinates are determined via those Laurent polynomials by specialising the $x_i$ to $a_i$ and $ \Delta_{K_j}$ to $a_j$ for $1 \leq i \leq N < j \leq N+n$.
Scott then only shows that this vector lies in $X$ and concludes that $\Phi_x$ is surjective. This confuses me as for that conclusion it would be left to show, that $v_a$ is an element in $U$ and $\Phi_x(v_a) = a$. I dont understand why that would be the case.
Link
[Sc] https://arxiv.org/pdf/math/0311148.pdf