Past exam of the short options course of the University of Oxford / 2023 / 3 (Projective geometry)

A projective conic in ${\mathbb{P}}^2(\mathbb{C})$ is the zero locus of a nonzero homogeneous quadratic polynomial, up to nonzero scaling of that polynomial. Under an invertible linear change of homogeneous coordinates, its symmetric matrix has rank $3$, $2$, or $1$. Over $\mathbb{C}$, the corresponding forms are a nonsingular conic such as $XZ-Y^2=0$, two distinct lines such as $XY=0$, or a double line $X^2=0$. In the last case the repeated factor records multiplicity, even though the underlying point set is one line.

A projective line is the projectivization of a two-dimensional linear subspace of ${\mathbb{C}}^3$, equivalently the zero locus of one nonzero linear form. For two such subspaces $U,W$, the dimension formula for a sum of subspaces gives $\dim (U\cap W)\ge 2+2-3=1$. A nonzero vector in the intersection therefore supplies a common projective point. Distinct lines have exactly one such point.

Use coefficient coordinates $[a:b:c]$ on $\mathbb{P}(V)$. A nonzero binary quadratic has a repeated linear factor precisely when its discriminant vanishes, soIts gradient is $(-4c,2b,-4a)$, which cannot vanish at a projective point. Therefore $C$ is a nonsingular projective conic. The parametrization $[\alpha :\beta ]\mapsto [{\alpha }^2:2\alpha \beta :{\beta }^2]$ identifies it as a degree-two Veronese map.

Assume $L\ne 0$. The quadratics divisible by $L$ form $L\cdot \mathrm{span}\{x,y\}$, a two-dimensional linear subspace; their projectivization is a projective line. Evaluating at the projective zero $[-\beta :\alpha ]$ of $L$ gives its equationAt the point $[L^2]=[{\alpha }^2:2\alpha \beta :{\beta }^2]$ of $C$, the tangent equation obtained from the gradient of $b^2-4ac$ is $-4{\beta }^{2}a+4\alpha \beta b-4{\alpha }^{2}c=0$. It is exactly the same line, establishing tangency.

The fundamental theorem of algebra factors $q(x,1)$ into linear factors over $\mathbb{C}$; homogenizing supplies factors of $y$ if the degree is less than three. Thus every nonzero homogeneous cubic is $c{L}_1{L}_2{L}_3$ for linear forms $L_i$, with $c\ne 0$ absorbed into one factor.

For pairwise nonproportional factors, part (c) assigns three distinct tangent lines ${\ell }_i=\mathbb{P}(L_i\cdot \mathrm{span}\{x,y\})$ to $C$. Their pairwise intersections are $[L_1{L}_2]$, $[L_1{L}_3]$, and $[L_2{L}_3]$. There cannot be a common point of all three: a nonzero quadratic cannot be divisible by three pairwise nonproportional linear factors, by unique factorization. Hence the three lines form a triangle circumscribed about $C$.