I dearly need help to proceed with the following exercise. Suppose I have the following map $$ g(x,y,z) = \begin{cases} [z,x] & \text{if } z \ne 0 \\ [0,y] & \text{if } z = 0 \end{cases} $$ from the set $$ y^2z = x(x-z)(x-2z) $$ to complex projective space. I would like to find its critical points and critical values, as well as the multiplicity at these values. The thing is I get an empty set of singular points, I expect otherwise so either I don't understand the task or I am doing something wrong. Denote $$ P(x,y,z) = y^2z - x(x-z)(x-2z) = y^2z - x^3+ 3zx^2 - 2xz^2\,. $$ The projective curve defined by $P(x,y,z)$ is $$ \mathcal{C} = \{[x,y,z] \colon P(x,y,z) = 0\} $$ Divide through by $z^3$ (for $z \neq 0$) and denote $X = \displaystyle \frac{x}{z}$, $Y = \displaystyle \frac{y}{z}$. Then we obtain $$ Y^2 = X(X-1)(X-2)\,, $$ hence set $$ g(x,y,z) = \tilde g(X,Y) = \begin{cases} [1,X] = [z,x] & \text{if } z \ne 0 \\ [0,1] = [0,y] & \text{if } z = 0 \end{cases} $$ EDIT: Now I can endow $\mathbb{C}$ with a complex coordinate by taking $X = x/z$ provided $z =\neq 0$, and $y$ if $z = 0$. (Is that correct?). How can I relate this to critical points ? Is it correct that the critical point is where $$ \frac{\partial P}{\partial Y} = 2Y = 0 $$ where $P(X,Y)$ is the polynomial given by the equation above? This would give me $$ y = z $$ but what does this tell me about the critical points ?? As pointed out kindly below in a comment below these should correspond to $X = 0,1$ or $2$ so $z^3 = 0$. What then are the critical values? Are the multiplicities all 1? I struggle to make sense of this task at ll (probably this is quite obvious here)..
Any hints would be helpful! (Is there a good reference for this? I tried Kirwan but I got lost, same for the Donaldson book.)