TITLE: Quotients by the additive group $\mathbb G_a$
QUESTION [7 upvotes]: Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a finitely generated invariant ring (e.g. when the additive group acts on a polynomial ring). 

How do I think of quotient varieties in this case?

I have a specific example in mind:
Let $G=\mathbb G_a$ act on the polynomial ring $k[x_0,\ldots,x_r,y_0,\ldots,y_r]$ by $$tx_i=x_i+ty_i, \text{ and } ty_i=y_i.$$ 
How should one think of a quotient $\mathbb P^{2r+1}/G$? 
Does it make sense to consider $\mathrm{Proj}$ of the invariant ring (i.e. the subring generated by $y_0,\ldots,y_r$ and the minors of the $2\times(r+1)$-matrix of variables)? 
If so, what does the resulting projective variety look like?

REPLY [3 votes]: The following response is not really an answer, as the second (and probably most pertinent) part of your question (the specific example) has already been addressed. 
However, with respect to your general question:

How do I think of quotient varieties in this case?

I would recommend:
Towards non-reductive geometric invariant theory by Doran and Kirwan.
They work out many good examples, and rather precisely address the case when the invariant ring is finitely generated.