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We define the variety $V$ as $\{ (x,y,z) \in \mathbb{A}^3\ |\ x = yz \}$. On this variety, I can make $(\mathbb{C}^*)^2$ act by $(\lambda, \mu) \star (x,y,z) = (\lambda \mu\ x , \lambda\ y,\mu\ z)$ on $V$.

How do you compute (=give an equation) the variety $V/(\mathbb{C}^*)^2$ ?

Thank you!

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1 Answer 1

up vote 1 down vote accepted

First of all, the projection $(x,y,z) \mapsto (y,z)$ is an isomorphism between $V$ and $\mathbb A^2$, so you may as well just consider the case $V =\mathbb A^2$, with $(\mathbb C^{\times})^{2}$ acting in the obvious way on the coordinates.

Then the open subset $U = \{ (y,z) \, | \, y z \neq 0\} \subset \mathbb A^2$ (i.e. the complement of the two coordinate lines) is acted on simply transitively by $(\mathbb C^{\times})^2$. Thus $U/(\mathbb C^{\times})^2$ is a point, and since $U$ is open and dense in $V$, to the extent that $V/(\mathbb C^{\times})^2$ has any meaning as a variety, it will be a point too.

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You're right when you say that $V/(\mathbb{C}^*)^2$ must already has a meaning as a variety, I was trying to simplify the definition of $V$ so that I could explain the issue more easily... But,anyway, thank you for your answer, I'll try to work it out in my situation then. –  ng_th Mar 15 '12 at 18:55

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