We consider geometric invariant theory for \emph{ graded additive groups}, groups of the form $\mathbb{ G}_a^r\rtimes_w\mathbb{ G}_m$ such that the $\mathbb{ G}_m$-action on $\mathbb{ G}_a^r$ is a scalar multiplication with weight $w\in\mathbb{ N}_+$. 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 $\mathbb{ G}_a^r$-quotient exists. This complements B\'erczi-Doran-Hawes-Kirwan, in the special case of one grading weight.
Categories: math.AG