## I.6: Nonsingular Curves

### 6.1. #to-work

Recall that a curve is rational if it is birationally equivalent to $\mathbf{P}^1(\mathrm{Ex} .4 .4)$. Let $Y$ be a nonsingular rational curve which is not isomorphic to $\mathbf{P}^1$.

(a) Show that $Y$ is isomorphic to an open subset of $\mathbf{A}^1$.
(b) Show that $Y$ is affine.
(c) Show that $A(Y)$ is a unique factorization domain.

### 6.2. An Elliptic Curve. #to-work

Let $Y$ be the curve $y^2=x^3-x$ in $\mathbf{A}^2$, and assume that the characteristic of the base field $k$ is $\neq 2$. In this exercise we will show that $Y$ is not a rational curve, and hence $K(Y)$ is not a pure transcendental extension of $k$.

(a) Show that $Y$ is nonsingular, and deduce that $A=A(Y) \simeq k[x, y] /\left(y^2-x^3+x\right)$ is an integrally closed domain.
(b) Let $k[x]$ be the subring of $K=K(Y)$ generated by the image of $x$ in $A$. Show that $k[x]$ is a polynomial ring, and that $A$ is the integral closure of $k[x]$ in $K$.
(c) Show that there is an automorphism $\sigma: A \rightarrow A$ which sends $y$ to $-y$ and leaves $x$ fixed. For any $a \in A$, define the norm of $a$ to be $N(a)=a \cdot \sigma(a)$. Show that $N(a) \in k[x], N(1)=1$, and $N(a b)=N(a) \cdot N(b)$ for any $a, b \in A$.
(d) Using the norm, show that the units in $A$ are precisely the nonzero elements of $k$. Show that $x$ and $y$ are irreducible elements of $A$. Show that $A$ is not a unique factorization domain.
(e) Prove that $Y$ is not a rational curve (Ex. 6.1). See (II, 8.20.3) and (III, Ex. 5.3) for other proofs of this important result.

### 6.3. #to-work

Show by example that the result of $(6.8)$ is false if either (a) $\operatorname{dim} X \geqslant 2$, or (b) $Y$ is not projective.

### 6.4. #to-work

Let $Y$ be a nonsingular projective curve. Show that every nonconstant rational function $f$ on $Y$ defines a surjective morphism $\varphi: Y \rightarrow \mathbf{P}^1$, and that for every $P \in \mathbf{P}^1$, $\varphi^{-1}(P)$ is a finite set of points.

### 6.5. #to-work

Let $X$ be a nonsingular projective curve. Suppose that $X$ is a (locally closed) subvariety of a variety $Y$ (Ex. 3.10). Show that $X$ is in fact a closed subset of $Y$. See (II, Ex. 4.4) for generalization.

### 6.6. Automorphisms of $\mathbf{P}^1$. #to-work 

Think of $\mathbf{P}^1$ as $\mathbf{A}^1 \cup\left\{{\infty}\right\}$. Then we define a fractional linear transformation of $\mathbf{P}^1$ by sending $x \mapsto(a x+b)/(c x+d)$, for $a, b, c, d \in k$, and $ad-b c \neq 0$.

(a) Show that a fractional linear transformation induces an automorphism of $\mathbf{P}^1$ (i.e., an isomorphism of $\mathbf{P}^1$ with itself). We denote the group of all these fractional linear transformations by $\PGL(1)$.
(b) Let Aut $\mathbf{P}^1$ denote the group of all automorphisms of $\mathbf{P}^1$. Show that Aut $\mathbf{P}^1 \simeq$ Aut $k(x)$, the group of $k$-automorphisms of the field $k(x)$.
(c) Now show that every automorphism of $k(x)$ is a fractional linear transformation, and deduce that $\mathrm{PGL}(1) \rightarrow$ Aut $\mathbf{P}^1$ is an isomorphism.

Note: We will see later (II. 7.1.1) that a similar result holds for $\mathbf{P}^n$ : every automorphism is given by a linear transformation of the homogeneous coordinates.

### 6.7. #to-work

Let $P_1, \ldots, P_r, Q_1, \ldots, Q_s$ be distinct points of $\mathbf{A}^1$. If $\mathbf{A}^1-\left\{P_1, \ldots, P_r\right\}$ is isomorphic to $\mathbf{A}^1-\left\{Q_1, \ldots, Q_{\diamond}\right\}$, show that $r=s$. Is the converse true? Cf. (Ex. 3.1).