I'm somewhat perplexed by the hint given in Exercise 1.27. The exercise itself says the following:
Look at the map: $\mu: [X_0,X_1]\mapsto [X_0^3,X_0X_1^2,X_1^3]=[Z_0,Z_1,Z_2]$. Show that the image for this map is the cubic hypersurface given by the equation $Z_0Z_2^2=Z_1^3$.
Hint: Show that the line $\alpha Z_1-\beta Z_2$ meets the curve defined by $Z_0Z_2^2=Z_1^3$ exactly at $[1,0,0]$ and $\mu([\beta,\alpha])$.
Now, showing that the image of $\mu$ is contained in the cubic hypersurface and that it meets the line in the aforementioned points is an easy task. What I do not understand, however, is the following:
What good did the hint do me? How do I know now that the hypersurface is the image of $\mu$?
Best Answer
Let me call the cubic hypersurface by $H$.
Now to see $H\subseteq \mu(\mathbb{C}P^1)$:
Let $P=[a_0, a_1, a_2]$ be a point on $H$. So that $a_0a_2^2=a_1^3$. If $a_0=0$ then $a_1=0$, so the point $P=[0: 0: 1]=\mu([0: 1])$.
So assume $a_0\neq 0$. So $P=[1: a_1: a_2]$ so that $a_2^2=a_1^3$.
If further $a_2=0$, then $a_1=0$, so $P=[1:0: 0]=\mu([1: 0])$, we are done in that case.
So let us assume that $a_2\neq 0$.
Consider the line $a_2Z_1-a_1Z_2$ passing through the point $P$, by the Hint it meets $H$ at $[1:0:0]$ and $\mu[a_1, a_2]$. Now check that $\mu[a_1, a_2]=P$.
To be honest, I haven't really (at least to my satisfaction) used the Hint in full force.