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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 $Proj$ 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?

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?

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 (i.e., 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 $Proj$ 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?

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 $Proj$ 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?

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 (i.e., 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 $Proj$ 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?

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 (i.e., 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 $Proj$ 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?