Finding projective transformations between curves

• May 9th 2012, 03:15 PM
Aliquantus
Finding projective transformations between curves
In the proof of the fact that after a projective transformation, the equation for any nonsingular cubic curve can be put in the form $y^2z=x(x-z)(x-\lambda z)$, where $\lambda \neq 0,1$, my notes in a middle-step arrives at the cubic

$(y-\frac{\alpha x+\beta z}{2})^2z=b_2(x,z)$,

and then the projective transformation

$[x,y,z]\mapsto [x,y-\frac{\alpha x+\beta z}{2},z]$

is applied to transform it to

$y^2z=b_3(x,z).$

I can't really see how this works - shouldn't we use the inverse of this projective transformation? To me it seems that replacing $y$ with $y-\frac{\alpha x+\beta z}{2}$ should give us the cubic $(y-\alpha x-\beta z)^2z=b_4(x,z)$, which is not really what we wanted.