Homework 2

Do any 5 of these problems:

1. Consider the quotient of the quasi-affine variety $\mathbb{A}^2\setminus\{0\}$, by the action of $k^\times$ , given by $\lambda\cdot(x,y)=(\lambda x,\frac{1}{\lambda}y)$.  Prove this quotient is a non-separated algebraic variety.
2. Consider the quotient of $\mathbb{A}^n$ by the symmetric group $S_n$, where the action is by permutation of the coordinates.  Prove that the quotient is isomorphic to $\mathbb{A}^n$. 
3. Prove that $\mathbb{A}^1\to\mathbb{A}^2, t\mapsto (t^2,t^3)$ has closed image, but is not a closed immersion (an isomorphism onto a closed subvariety). 
4. Prove that all automorphisms of $\mathbb{P}^1$ are linear fractional transformations $x\mapsto \frac{ax+b}{cx+d}$ (in an affine coordinate).  The automorphism group of $\mathbb{P}^1$ acts simply transitively on triples of distinct points.
5. Given two non-intersecting lines $L_1, L_2 \subset \mathbb{P}^3$, and a point $P$ not on either line, there exists a unique line through $P$ intersecting both $L_1$ and $L_2$. 
6. If $n>m$, then every morphism $\mathbb{P}^n\to\mathbb{P}^m$ is constant.
7. Prove that $\mathbb{P}^n\times\mathbb{P}^m$ is not isomorphic to $\mathbb{P}^{n+m}$.