I was trying some exercises from Silverman's book Rational Points on Elliptic Curves 2nd ed. (2015), and got stuck at this problem.
1.22 Let $C$ and $W$ be the projective curves ($b,e \ne 0$) $$ C: XY^2 + (aX+bZ)YZ = cX^2Z + dXZ^2 + eZ^3 $$ $$ W: Y^2Z + (aX+bZ)YZ = cX^3 + dX^2Z + eXZ^2 $$ Consider points on $C$, $$ \mathcal{O}=[1,0,0], \: P =[0,1,0], \: Q = [0,e,b] $$ and points on $W$, $$ \mathcal{O'}=[0,1,0], \: P' =[0,-b,1], \: Q' = [0,0,1] $$ then there exists a map from $C$ to $W$. Show that under this map $\mathcal{O},P,Q$ are sent to $\mathcal{O'},P',Q'$. Moreover, $\mathcal{O},P,Q$ are non-singular points on $C$ if and only if $\mathcal{O'},P',Q'$ are non-singular points on $W$ respectively (The original exercise is longer than this but this is the part I have trouble with).
In the section where he derives the Weierstrass normal form, there is a key step to multiply both sides by $x$ and replace $xy$ by $y$. So I did the analogue, when $Z \ne 0$, multiply both sides by $X/Z$ and replace $XY/Z$ by $Y$ to go from $C$ to $W$. Now I am asked to establish correspondance between the points under the map, but I have no idea how points at infinity should be mapped to under the map as I don't even know if it is allowed to use the same map when $Z = 0$. It wouldn't make much sense to substitute $Y' = XY/Z$ in the case. My first question is that is the map $X \to X'=X$, $Y \to Y' = XY/Z$, $Z \to Z'$ even correct?
Now I have to establish the non-singular point, I compute $$ \frac{\partial C}{\partial X} = Y^2 +aYZ -2cZX - dZ^2, \frac{\partial C}{\partial Y} = 2XY +(aX+bZ)Z, \\ \frac{\partial C}{\partial Z} = aXY + 2bYZ -cX^2 -2dXZ -3eZ^2 $$ and $$ \frac{\partial W}{\partial X} = aYZ -3cX^2 -2dZX -eZ^2, \frac{\partial W}{\partial Y} = 2ZY + (aX+bZ)Z, \\ \frac{\partial W}{\partial Z} = Y^2 +aXY +2bYZ - dX^2 -2eXZ $$ If $\mathcal{O}$ is a singular point, then $c=0$. (First two partial derivatives evaluate to $0$). But $\dfrac{\partial W}{\partial Z} = 1 \ne 0$ for $\mathcal{O'}$, so it is never singular. They do not match at all.
What has gone wrong there? I can't seem to figure it out.
EDIT: The first edition of the question, which is differently formulated than the one appearing in the second edition, can be found on page 3 of the errata list http://www.math.brown.edu/~jhs/RPEC/RPECErrata.pdf.