Help in this very basic example in algebraic curves

Question

I'm trying to understand this example:

I didn't understand why the second factor describes a point of intersection $q$, since the second factor doesn't vanish at $q$.

Anyone can clarifies this for me please?

Thanks

Answer 1

Consider the equation $\lambda_1^2T_0-\lambda_0^3T_1=0$. Solving gives $$\frac{T_0}{T_1} = \frac{\lambda_0^3}{\lambda_1^2}$$ In other words $(T_0:T_1) = (\lambda_0^3:\lambda_1^2)$. Putting this into $X(T)$ gives $$(T_0:\lambda_0T_1:\lambda_1T_1)=(\lambda_0^3:\lambda_0\lambda_1^2:\lambda_1^3)$$

Answer 2

The equation $T_1^2 (\lambda_1^2 T_0 - \lambda_0^3 T_1) = 0$ describes the curve in terms of its parameterization in ${\mathbb P}_1$ rather than as its image in ${\mathbb P}_2$. So you should evaluate $\lambda_1^2 T_0 - \lambda_0^3 T_1$ at $(\lambda_0^3, \lambda_1^2)$ (which corresponds to $q = (\lambda_0^3, \lambda_0\lambda_1^2, \lambda_1^3$)); that gives $0$.