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Consider the following $G = (\mathbb{C}^*)^{\times 3}$-action on $\mathbb{A}^6$:

$(\lambda_0,\lambda_1,\lambda_2) \cdot (x_0,x_1,x_2,y_0,y_1,y_2) = (\lambda_0x_0,\lambda_1x_1,\lambda_2x_2,\frac{\lambda_0^2}{ \lambda_1\lambda_2}y_0,\frac{\lambda_1^2}{ \lambda_0\lambda_2}y_1,\frac{\lambda_2^2}{\lambda_0\lambda_1}y_2)$.

I need some help determining the GIT quotient of $\mathbb{A}^6$ by $G$.

The union of zero-sets $Z = V(x_1,y_1) \cup V(x_2,y_2) \cup V(x_3,y_3) \cup V(x_1,x_2,x_3)$ in $\mathbb{A}^6$ appears to be the locus of non-stable points.

How does the resulting geometric quotient of $\mathbb{A}^6 \backslash Z$ by $G$ look like as an algebraic variety? How does one generally go about in determining such quotients after identifying the stable locus?

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    $\begingroup$ What sort of answer would be acceptable for "what does […] look like as an algebraic variety"? $\endgroup$ – LSpice Sep 5 at 22:31
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    $\begingroup$ I get that my question may be somewhat ambiguous, but I am looking for a description of a variety which does not have (explicit or implicit) reference to the particular action. Preferably a description would be an explicit $G$-invariant map to affine or projective space whose image realizes the quotient. Or perhaps an identification with something more familiar, such as a toric variety (with its fan), as a blow-up of $\mathbb{P}^3$, a succinct equation, etc.. $\endgroup$ – Mellon Sep 5 at 22:39
  • $\begingroup$ I should add: the $G$-invariant map should have a target of minimal dimension. $\endgroup$ – Mellon Sep 6 at 8:41

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