TITLE: Hartshorne exercise 1.1 (c)
QUESTION [12 upvotes]: I want to know how can I find the conditions where $A(W)$, the affine coordinate ring of a variety given by an irreducible quadratic polynomial in $k[x,y]$, is isomorphic to $A(V)$ or to $A(Z)$ where $V$ is a parabola defined by $y=x^2$ and $Z$ is the hyperbola given by $xy=1$. Thanks in advance for your answers.

REPLY [15 votes]: Here's a solution that'll work for all characteristics - Factorize the degree $2$ homogeneous part into linear factors (can do this because algebraically closed). Now, if the linear factors are linearly dependent, w.l.o.g. change coordinates to make this linear factor the new $X$. The equation now becomes $X^2 + aX + bY + c$. By now making $aX+bY+c$ the new $Y$ ($b \neq 0$), we get $X^2+Y$.
If the factors are linearly independent, change coordinates to make one of them $X$ and the other $Y$. The equation now becomes $XY + aX + bY + c$ which we can write as $(X+b)(Y+a)+d$. We can now change coordinates to make it $XY+d$ and so, we're done.
As an interesting aside, this indicates as expected that the general equation is a hyperbola since it corresponds to the linearly independent case here.