## I.4: Rational Maps

### 4.1. #to-work

If $f$ and $g$ are regular functions on open subsets $U$ and $V$ of a variety $X$, and if $f=g$ on $U \cap V$. show that the function which is $f$ on $U$ and $g$ on $V$ is a regular function on $U \cup V$. Conclude that if $f$ is a rational function on $X$, then there is a largest open subset $U$ of $X$ on which $f$ is represented by a regular function. We say that $f$ is defined at the points of $U$.

### 4.2. Same problem for rational maps. #to-work

If $\varphi$ is a rational map of $X$ to $Y$, show there is a largest open set on which $\varphi$ is represented by a morphism. We say the rational map is defined at the points of that open set.

### 4.3. #to-work

1.  Let $f$ be the rational function on ${\mathbb{P}}^2$ given by $f=x_1 / x_0$. Find the set of points where $f$ is defined and describe the corresponding regular function.
2.  Now think of this function as a rational map from ${\mathbb{P}}^2$ to $\mathbf{A}^1$. Embed $\mathbf{A}^1$ in ${\mathbb{P}}^1$, and let $\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^1$ be the resulting rational map. Find the set of points where $\varphi$ is defined, and describe the corresponding morphism.

### 4.4. #to-work

A variety $Y$ is rational if it is birationally equivalent to ${\mathbb{P}}^n$ for some $n$ (or, equivalently by (4.5), if $K(Y)$ is a pure transcendental extension of $k$ ).

1.  Any conic in ${\mathbb{P}}^2$ is a rational curve.
2.  The cuspidal cubic $y^2=x^3$ is a rational curve.
3.  Let $Y$ be the nodal cubic curve $y^2 z=x^2(x+z)$ in ${\mathbb{P}}^2$. Show that the projection $\varphi$ from the point $P=(0,0,1)$ to the line $z=0$ (Ex. 3.14) induces a birational map from $Y$ to ${\mathbb{P}}^1$. Thus $Y$ is a rational curve.

### 4.5. #to-work

Show that the quadric surface $Q: x y=z w$ in ${\mathbb{P}}^3$ is birational to ${\mathbb{P}}^2$, but not isomorphic to ${\mathbb{P}}^2$ (cf. Ex. 2.15).

### 4.6. Plane Cremona Transformations. #to-work

A birational map of ${\mathbb{P}}^2$ into itself is called a plane Cremona transformation. We give an example, called a quadratic transformation. It is the rational map $\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^2$ given by $\left(a_0, a_1, a_2\right) \rightarrow\left(a_1 a_2, a_0 a_2, a_0 a_1\right)$ when no two of $a_0, a_1, a_2$ are $0$.

1.  Show that $\varphi$ is birational, and is its own inverse.
2.  Find open sets $U, V \subseteq {\mathbb{P}}^2$ such that $\varphi: U \rightarrow V$ is an isomorphism.
3.  Find the open sets where $\varphi$ and $\varphi^{-1}$ are defined. and describe the corresponding morphisms. See also (Chapter V, 4.2.3).

### 4.7. #to-work

Let $X$ and $Y$ be two varieties. Suppose there are points $P \in X$ and $Q \in Y$ such that the local rings ${\mathcal{O}}_{P, X}$ and ${\mathcal{O}}_{Q, Y}$ are isomorphic as $k{\hbox{-}}$algebras. Then show that there are open sets $P \in U \subseteq X$ and $Q \in V \subseteq Y$ and an isomorphism of $U$ to $V$ which sends $P$ to $Q$.

### 4.8. #to-work

1.  Show that any variety of positive dimension over $k$ has the same cardinality as $k$.[^1]
2.  Deduce that any two curves over $k$ are homeomorphic (cf. Ex. 3.1).

### 4.9. #to-work

Let $X$ be a projective variety of dimension $r$ in $P^n$. with $n \geqslant r+2$. Show that for suitable choice of $P \notin X$. and a linear ${\mathbb{P}}^{n-1} \subseteq {\mathbb{P}}^n$. the projection from $P$ to ${\mathbb{P}}^{n-1}$ (Ex. 3.14) induces a birational morphism of $X$ onto its image $X' \subseteq {\mathbb{P}}^{n-1}$. You will need to use (4.6A). (4.7A). and (4.8A). This shows in particular that the birational map of (4.9) can be obtained by a finite number of such projections.

### 4.10. #to-work

Let $Y$ be the cuspidal cubic curve $y^2=x^{3}$ in $\mathbf{A}^2$. Blow up the point $O=(0.0)$. Let $E$ be the exceptional curve. and let $\tilde{Y}$ be the strict transform of $Y$. Show that $E$ meets $\tilde{Y}$ in one point. and that $\tilde{Y} \cong \mathbf{A}^1$. In this case the morphism $\rho: \tilde{Y} \rightarrow Y$ is bijective and bicontinuous. but it is not an isomorphism.

[^1]: Hint: Do $\mathbf{A}^n$ and ${\mathbb{P}}^n$ first. Then for any $X$, use induction on the dimension n. Use (4.9) to make $\mathrm{X}$ birational to a hypersurface $H \subseteq {\mathbb{P}}^{n+1}$. Use (Ex. 3.7) to show that the projection of $H$ to ${\mathbb{P}}^n$ from a point not on $H$ is finite-to-one and surjective.