We consider geometric invariant theory for graded additive groups, groups of the form GₐʳwGₘ such that the Gₘ-action on Gₐʳ is a scalar multiplication with weight w₊. We provide an algorithm of equivariant birational modifications, such that we can apply the geometric invariant theory of B\'erczi-Doran-Hawes-Kirwan. In particular, the geometric Gₐʳ-quotient exists. This complements B\'erczi-Doran-Hawes-Kirwan, in the special case of one grading weight.
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Yikun Qiao (2024) studied this question.
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