(a) Define what it means to give a rational map between algebraic varieties. Define a birational map.

(b) Let

$$X=Z\left(y^{2}-x^{2}(x-1)\right) \subseteq \mathbb{A}^{2}$$

Define a birational map from $X$ to $\mathbb{A}^{1}$. [Hint: Consider lines through the origin.]

(c) Let $Y \subseteq \mathbb{A}^{3}$ be the surface given by the equation

$$x_{1}^{2} x_{2}+x_{2}^{2} x_{3}+x_{3}^{2} x_{1}=0 .$$

Consider the blow-up $X \subseteq \mathbb{A}^{3} \times \mathbb{P}^{2}$ of $\mathbb{A}^{3}$ at the origin, i.e. the subvariety of $\mathbb{A}^{3} \times \mathbb{P}^{2}$ defined by the equations $x_{i} y_{j}=x_{j} y_{i}$ for $1 \leqslant i<j \leqslant 3$, with $y_{1}, y_{2}, y_{3}$ coordinates on $\mathbb{P}^{2}$. Let $\varphi: X \rightarrow \mathbb{A}^{3}$ be the projection and $E=\varphi^{-1}(0)$. Recall that the proper transform $\tilde{Y}$ of $Y$ is the closure of $\varphi^{-1}(Y) \backslash E$ in $X$. Give equations for $\tilde{Y}$, and describe the fibres of the morphism $\left.\varphi\right|_{\widetilde{Y}}: \widetilde{Y} \rightarrow Y$.