0

Suppose I have a projective variety $X$ (for which I have explicit equations) and an involution $\iota$ on it (again, explicit). I'd like to write down explicit equations for $X/\langle \iota \rangle$, but I'm not sure how to proceed (I've seen the theoretical construction, but it didn't help me much in the task of writing down explicit equations).

For the sake of an example, say, I have a projective variety $X=V(ax^2+bxy+cy^2−z^2)⊆\mathbb{P}^2$ and I want to quotient this by the involution $ι:[x:y:z]↦[x:y:−z]$. Can someone explain to me how to get the explicit equations of $X/⟨ι⟩$? If this is not a good example, can someone provide a better example?

user131642
  • 1
  • To have equations for $X / \langle \iota\rangle$ you have to have an embedding of it in some projective space. Once you quotient $X \subset \mathbb P^2$ you normally leave the projective space you started with, so your question is a little problematic. – Gunnar Þór Magnússon Feb 26 '14 at 16:09

1 Answers1

0

(An answer to the original question, but not the edited one.) A quick and dirty argument is just to observe that the quotient map identifies points if and only if they lie on the same line through $[0,0,1]$. So the quotient map is the same as the projection from that point onto $\mathbf P^1$. That is, the quotient is defined by the empty set of equations in $\mathbf P^1_{[x,y]}$.

By the way, it's a little odd to say "suppose for simplicity that this involution is fixed-point free", when it is definitely not fixed point free (over an algebraically closed field at least).

  • Can you give me the full argument explicitly? The example above is just a toy example, I want to be able to do these kind of construction in general. To complicate things a bit more, suppose $Y = X \cap \mathbb{V}(\alpha x^2 + \beta y^2 + t^2) \subseteq \mathbb{P}^3$ and the involution is now $\iota : [x:y:z:t] \mapsto [x:y:-z:-t]$. How to get the explicit equations of $Y / \langle \iota \rangle$? – user131642 Feb 26 '14 at 15:03
  • Dear @user131642, you can see Matt E's detailed answer here: http://math.stackexchange.com/questions/66378/quotient-varietiesrq=1 for an explanation of how to deal with the general case. –  Feb 26 '14 at 15:08
  • 1
    Let me add that in general, it is a good idea to be clear in your question about what you really want to know. It is somewhat discouraging to provide an answer, only for the OP to respond "Yes, but I don't really care about that; what I really want to know is..." –  Feb 26 '14 at 15:10
  • Thanks. But I'd seen Matt E's answer, and still feel I don't know how to get the equations for explicitly. Do you have an easy (but non-trivial) explicit example which might help my understanding? – user131642 Feb 26 '14 at 15:12
  • Dear Asal, I'm new to SE, sorry for the misleading formulation of the question - I'll change it into a more precise one. – user131642 Feb 26 '14 at 15:13
  • Unfortunately, I don't know that there's a straightforward general way to find equations for the quotient variety. (Note that projective varieties don't always come with a fixed projective embedding --- quotients being a prime example --- so one cannot really speak of "the" equations of the quotient.) But maybe someone else can say something more helpful. –  Feb 26 '14 at 15:23