# Intersection of projective quadric and affine plane.

I'm stuck in trying to understand the graphical part of the following problem.

Let $$\mathcal C = \{ [x:y:w:z] \in \mathbb P ^3: x^2 +xy +yw -w^2 = 0 \}$$. Graph the intersection $$\mathcal C \cap \mathbb A ^3$$.

What I have tried is the following. I found the canonical projective form of the quadric which if I'm not mistaken is: $$\mathcal C(x^2 - y^2)$$. The notation means the quadric generated by that form. Now I know that $$\mathbb A ^3$$ are the points where $$x \neq 0$$. So if everything is correct I'm left with the following points in $$\mathbb P ^3 \cap \mathbb A ^3$$.

$$[x,x,w,z], x\neq 0 \hspace0.75cm \text{and} \hspace0.75cm [x,-x,w,z], x\neq 0 .$$

Are these plane (two dimensionals as $$z$$ and $$w$$ are free) where the $$y$$ value is fixed at either $$1$$ or $$-1$$? I'm following the idea that $$\mathbb A ^3$$ is an hyperplane in $$\mathbb A^4$$ where the first coordinate is fixed at 1 as it's different from 0.

Sorry if it's not clear enough as I'm still wrapping my head around these concepts.

Instead of merely going for $$x\neq 0$$ you might as well pick one specific representative and assume $$x=1$$. So you are looking for points $$(y,w,z)\in\mathbb A^3$$ with $$y+yw-w^2=0$$. This is independent of $$z$$, so you can plot this as a planar curve and then extrude it along the $$z$$ axis. To graph the planar curve, a hyperbola, you can compute asymptotes and probably a few values, then judge a reasonable interpolation in between. Unless you have learned different techniques.
• Sorry for not including the details of the canonical form. Yes, I applied several projective transformations to get to that. So, the intersection in this case actually corresponds to the points $y + yw - w^2 = 0$ so that means you are left with a curve in the $(y,w)$ plane and as $z$ is free, you are left with a surface in $\mathbb A ^3$. Is this correct? – Leo Sep 23 '18 at 22:16
• @LeoLerena $\mathcal C$ factorizes into $(x + w) (x + y - w)$. – Maxim Nov 7 '18 at 20:52