# Lecture 13

Note If a conic in $\mathbb{R}P^{2}$ contains a line, then it is degenerate.

Theorem. Through five points $P_{1},P_{2},P_{3},P_{4},P_{5} \in \mathbb{R}P^{2}$ there exists a conic in $\mathbb{R}P^{2}$.

$i)$ If no four points lie on a line, then the conic is unique.

$ii)$ If no three lie on one line (five points in general position), then the conic is non-degenerate.

Lemma. If a conic contains three collinear points, then it contains the line through these points. Continue reading

# Lecture 12

Definition.
Let $V$ be a vector space over $\mathbb{R}$ (or $\mathbb{C}$). A map
$b\colon V \times V \to \mathbb{R}$ is a symmetric bilinear form, if

1. $b(v,w)=b(w,v) \quad \forall v,w \in V$
2. $b(\alpha_1 v_1 + \alpha_2 v_2,w) = \alpha_1 b(v_1,w) + \alpha_2 b(v_2,w)$ for all $\alpha_1,\alpha_2 \in \mathbb{R}$, $v_1, v_2, w \in V$.

$b$ is non-degenerate, if
$b(v,w)=0 \quad \forall w \in V \Rightarrow v=0\,.$
The corresponding quadratic form is defined by
$q(v)=b(v,v) \quad \forall v \in V\,.$

# Exercise sheet 5

In exercise 2 on sheet 5, the claim to be proven ($\textrm{cr}(y,p,x,q) = -1$) of course refers to the dual construction. (In the primal construction, these 4 lines are not concurrent).

# Emanuel’s office hours

Please participate in the doodle poll about changing Emanuel’s office hours to Monday.

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According to the poll, Emanuel’s office hours have been changed to Monday, 10:30 – 12, starting from Monday the 26th on. There is also the option to visit Thilo’s office hours on Tuesdays, 13-14.

# 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

# Exercise sheet 4

In exercise 3 about the fundamental theorem of affine geometry, it has to be $n > 1$. Otherwise, the statement is obviously false, as for the fundamental theorem of projective geometry.
(The exercise sheet has been updated accordingly)

# Lecture 9

Fundamental Theorem of real projective geometry: Let $f\colon \RP^n \to \RP^n$, $n \ge 2$, be a bijective map that maps lines to lines. Then $f$ is a projective transformation.

Remark: The theorem does not hold for arbitrary fields. For example $f\colon \CP^n \to \CP^n$ ($n \ge 2$) with
$f\left(\sqvector{z_1\\ \vdots\\ z_{n+1}}\right) = \sqvector{\bar z_1\\ \vdots\\ \bar z_{n+1}}$
is a bijective map, mapping complex projective lines to complex projective lines but it is not a projective transformation of $\CP^n$.

# Lecture 8

Theorem (complete quadrilateral). Consider four lines with intersection points $A$, $B$, $C$, $D$, $P$, and $Q$ as shown in the picture. Then the cross-ratio of the intersection points $X$ and $Y$ of the diagonals with the line $PQ$ and $P$ and $Q$ is:

$\cr(P,X,Q,Y)=−1.$

Proof of theorem on complete quadrilateral (multi-ratio).
Consider the multi-ratio with

$P_{1} = Q_{1} = P$, $P_{3} = Q_{3} = Q$, $Q_{2} = X$, $P_{2} = Y$,

where $p$ is the affine coordinates of $P$, $q$ the ones of $Q$ etc.

\begin{align}
-1 &= \mathrm{m}(P, X, Q, P, Y, Q) \\
&= \frac{p-x}{x-q} \frac{q-p}{p-y} \frac{y-q}{q-p} \\
&= \frac{p-x}{x-q} \frac{q-y}{y-p} \\
&= \mathrm{cr}(P, X, Q, Y). \\
\end{align}

$\square$