# Lecture 11

Cross-ratio of lines through a point (again).

Claim. The map which maps a point $g^{\ast} \in p^{\ast}$ to the intersection point of $g \cap \ell$ is a projective map. Continue reading

# Server problems

We had some server problems and are working on it. Sorry for the inconvenience.

# Lecture 7

Definition (complete quadrilateral). A configuration consisting of four lines in the projective plane – no three through one point – and the six intersection points, one for each pair of lines, form a complete quadrilateral.

# Room changes

Sorry for the inconvenience today. The other dates that we have to leave the regular room are:

Monday 10.12. MA 649

Thursday 13.12. MA 650

# Notes as single pdf

A pdf version of the notes is maintained by Che Netzer and available for download on his website.

# Projective transformations

Let $V$, $W$ be two vectorspaces over the same field and of the same dimension and $F\colon V \rightarrow W$ a linear isomorphism. In particular $ker(F) = \{0\}$, so F maps 1-dimensional subspaces to 1-dimensional subspaces.

Hence $F$ induces a map from $P(V)$ to $P(W)$.

Definition: A projective transformation $f$ from $P(V)$ to $P(W)$ is a map defined by a linear isomorphism $F\colon V \rightarrow W$ such that

\begin{equation*}
f([v]) = [F(v)] \quad \forall [v] \in P(V)\,.

# Desargues and Pappus configuration

I put the configurations I showed on the projector online:

You may move the points around to see how the configuration changes.

# Q: Points in general position

Question. The above picture only show an affine part of the respective projective spaces. What happens if some of the points lie at infinity?