- Write down the dual of Pappus' Theorem.
- Let $P_{0}, P_{1}, P_{2}, P_{3}$ be four distinct points in a projective plane $\mathbb{P}(V)$. Show that $P_{0}, P_{1}, P_{2}, P_{3}$ are in general position if and only if the lines $P_{0} P_{1}, P_{1} P_{2}, P_{2} P_{3}, P_{3} P_{0}$ are in general position in $\mathbb{P}\left(V^{*}\right)$.
- Use general position arguments to show that given five points in the projective plane, such that no three are collinear, there is a unique conic through these five points.
- Let $C, D$ be conics in a projective plane $\mathbb{P}(V)$, where $V$ is a 3-dimensional real vector space, and suppose that $C \cap D=\left\{p_{1}, p_{2}, p_{3}, p_{4}\right\}$, where $p_{1}, \ldots, p_{4}$ are distinct points in $\mathbb{P}(V)$. (a) Show that $p_{1}, \ldots, p_{4}$ are in general position. Prove that there exist homogeneous coordinates $\left[x_{0}: x_{1}: x_{2}\right]$ on $\mathbb{P}(V)$ for which $$ p_{1}=[1: 1: 1], \quad p_{2}=[1:-1: 1], \quad p_{3}=[1: 1:-1], \quad p_{4}=[1:-1:-1] . $$ (b) Show that any conic through $p_{1}, \ldots, p_{4}$ has equation $$ \lambda x_{0}^{2}+\mu x_{1}^{2}+\nu x_{2}^{2}=0 $$ where $\lambda+\mu+\nu=0$ (c) Find four projective transformations $\tau$ of $\mathbb{P}(V)$ that form a group, and for which $\tau(C)=C$ and $\tau(D)=D$.
- Let $F\left(x_{0}, x_{1}, x_{2}\right)$ be a homogencous polynomial of degree $n$. Let $\mathcal{C}$ be the set of points $\left[a_{0}, a_{1}, a_{2}\right]$ in $\mathbb{R} \mathbb{P}^{2}$ such that $F\left(a_{0}, a_{1}, a_{2}\right)=0$. Let a be a point on $\mathcal{C}$. Provided that $\nabla F(\mathbf{a}) \neq \mathbf{0}$, the tangent line to $\mathcal{C}$ at a $=\left[a_{0}, a_{1}, a_{2}\right]$ is the line $$ x_{0} \frac{\partial F}{\partial x_{0}}(\mathbf{a})+x_{1} \frac{\partial F}{\partial x_{1}}(\mathbf{a})+x_{2} \frac{\partial F}{\partial x_{2}}(\mathbf{a})=0 $$ in $\mathbb{R P}^{2}$ and a is said to be singular if $\nabla F(\mathbf{a})=\mathbf{0}$. (i) Show that a lies on the tangent line to a. (ii) Given a $3 \times 3$ symmetric real matrix $B$ its associated conic is the set of solutions to the equation $\mathrm{x}^{T} B \mathrm{x}^{T}=0$ where $\mathrm{x}=\left[x_{0}: x_{1}: x_{2}\right]$ and the conic is said to be singular if $B$ is singular. Show that a conic is singular if and only if it has a singular point. (iii) Sketch the curves $y^{2}=x^{3}$ and $y^{2}=x^{2}(x+1)$ in $\mathbb{R}^{2}$. What singular points do these curves have? Show that $y=x^{3}$ has a singular point at infinity.
- Find all rational numbers $x, y$ such that $x^2+y^2-x y=1$.
- Let $V$ be a 3-dimensional real vector space and suppose that $L_{0}, L_{1}, L_{2}, L_{3}$ are four lines in the projective plane $\mathbb{P}(V)$ all intersecting in a common point $x$. Explain why (i) if $L$ is a line in $\mathbb{P}(V)$ that does not pass though $x$, but intersects $L_{i}$ in a point $x_{i}$ (so $x_{0}, x_{1}, x_{2}, x_{3}$ are four distinct collinear points), then the cross-ratio $\left(x_{0} x_{1}: x_{2} x_{3}\right)$ is independent of the choice of $L$; (ii) the cross-ratio defined in (i) equals the cross-ratio $\left(L_{0} L_{1}: L_{2} L_{3}\right)$ formed by regarding $L_{0}, L_{1}, L_{2}, L_{3}$ as collinear points of the dual projective plane $\mathbb{P}\left(V^{*}\right)$.