Okay, so do you know if the exponential to $G_0$ is open in general?

Thanks! By locally surjective I mean that the image of every nonempty open set has nonempty interior. Not sure whether this implies the map being open.

Ah, yes! Thanks! Do you know if we can at least say that it is locally injective/bijective/surjective (which seems to be the case in your example)?

I am sorry if this was confusing. I thought that these things were kind of equivalent because you can view the finitely generated module over $A$ also as a $K[x_1,\ldots,x_n]$ where $x_1,\ldots,x_n$ are mapped to generators of $A$.

@DanielAsimov I work over the complex numbers and I am looking at the Grassmannian of lines in $\mathbb{P}^4$. Anyways, I think I can work with what Jason Starr wrote (thanks Jason!)