Math 480, Spring 2014, Assignment 12

Carefully define the following terms, then give one example and one non-example of each:

1. Regular map (from $\mathbb{P}^n$ to $\mathbb{P}^m$).
2. Degree (of a regular map).
3. Projective transformation (a.k.a. automorphism of $\mathbb{P}^n$).
4. Congruent (subsets of $\mathbb{P}^n$).
5. General position (a property of $k$-tuples of points in $\mathbb{P}^n$).

Carefully state the following theorems (you need not prove them):

1. Theorem describing the group $\mathrm{Aut}(\mathbb{P}^n)$.
2. Theorem on congruence of tuples in general position.
3. Theorem on congruence of $d$-planes.

Solve the following problems:

1. Let $H\subset\mathbb{P}^n$ be any hyperplane (i.e. any $(n-1)$-plane) and let $p\in\mathbb{P}^n$ be any point not lying on $H$. Show that there is a projective transformation $\phi\in\mathrm{Aut}(\mathbb{P}^n)$ such that $\phi(p)$ is the "origin" $(1,0,\dots,0)$ and $\phi[H]$ is the "line at infinity" $x_0=0$.
2. (Projection from a point to a hyperplane) Let $p$ and $H$ be as above. Define a function $\pi_{p,H}:\mathbb{P}^n-\{p\}\rightarrow H$ by letting $\pi_{p,H}(x)$ be the point in which the line through $p$ and $x$ intersects $H$. Taking $n=2$ and $\mathsf{k}=\mathbb{R}$, draw two pictures illustrating the action of $\pi_{p,H}$: one in which $p=(1,0,0)$ and $H$ is the line $x_1=1$, and one in which $p=(1,0,0)$ and $H$ is the line at infinity $x_0=0$.
3. Show that the map $\pi_{p,H}$ defined above is "locally regular," i.e. there exist homogeneous polynomials $p_0,\dots,p_n$, all of the same degree, such that $\pi_{p,H}(x_0,\dots,x_n)=(p_0,\dots,p_n)$. (Note that $\pi_{p,H}$ is not globally regular -- it is not even globally well-defined -- since the polynomials turn out to vanish simultaneously at $p$.) (Hint: by the result of the first problem, it suffices to prove this in the case where $p$ is the origin and $H$ is the hyperplane at infinity. Referring to your second sketch above, you may find it straightforward to write an explicit formula for $\pi_{p,H}$ in this case.)