Let $S$ be a irreducible scheme over a field $k$ (for example a smooth projective curve over algebraically closed field). Denote by $k(S)$ its field of fractions. Let $K$ be a(n algebraically closed) field. Given an embedding $k(S) \hookrightarrow K$ we can construct a $K$-point of $S\otimes K$ in the following way: choose an affine open set in $S$ with the coordinate ring $R$. Then we can restrict the embedding to the embedding $R \hookrightarrow K$. After tensor product with $K$ it becomes a $K$-point in $S\otimes K$.

My question is: which points arise as the image of this map?