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Definition

Let $k$ be a field. A conic section over $k$ is the zero set of a degree 2 polynomial $p(x, y) \in k[x, y]$ in the affine plane $\mathbb{A}^2(k)$, or better yet the zero set of a homogeneous polynomial $p(x, y, z)$ of degree 2 in the projective plane $\mathbb{P}^2(k)$.

Over the real numbers

In the classical case $k = \mathbb{R}$ of real numbers, conic sections may be pictured in terms of intersections of a standard cone $x^2 + y^2 - z^2 = 0$ in affine 3-space with various affine hyperplanes (hence the name, “conic section”). In this picture, nonsingular conic sections are classified (up to automorphisms of the affine plane) by the sign of the discriminant of $p$. In other words, if we write $p(x, y) = a x^2 + b x y + c y^2 + \lambda(x, y)$ where $deg(\lambda) = 1$ and put $D = b^2 - 4 a c$, then an isomorphism class is one of three types: ellipses (when $D \lt 0$), parabolas ($D = 0$), and hyperbolas ($D \gt 0$). Of course when we admit possibly singular conic sections, we get further isomorphism classes involving some degree of degeneracy (e.g., two lines, a double line, etc.).

The distinctions between ellipse, parabola, and hyperbola are artifacts of affine geometry: if we instead consider conic sections as projective subvarieties of $\mathbb{P}^2(\mathbb{R})$, then considered up to projective transformations (automorphisms of the projective plane), these distinctions evaporate and there is really only one kind of nonsingular conic section. Put differently: if we fix a representation $\mathbb{P}^2(\mathbb{R}) = \mathbb{A}^2(\mathbb{R}) \sqcup L$ where $L$ is a chosen “line at infinity”, then in the original classification up to affine transformations, i.e., the subgroup of projective transformations which take $L$ to itself, ellipses are those conic sections which do not intersect $L$, parabolas are those which intersect $L$ in a double point, and hyperbolas are those which intersect $L$ in two points. By enlarging to the group of all projective transformations, we can move $L$ to a line which does intersect an “ellipse” in two points, making it a “hyperbola” with respect to the new coordinate system, etc.

Stereographic projection

Considered in terms of projective geometry, all pointed1 nonsingular conic sections $C \subset \mathbb{P}^2(k)$ are isomorphic and can be identified explicitly with a projective line $\mathbb{P}^1(k)$ by means of a stereographic projection.

Geometrically, if $p$ is the chosen basepoint of $C$ and $L \subset \mathbb{P}^2(k)$ is a line not incident to $p$, then for any other point $q$ of $C$ the unique line $L(p, q)$ incident to $p$ and $q$ intersects $L$ in exactly one point, denoted $\phi(q)$. (Here $\phi(p)$ to be the intersection of the tangent to $p$ at $C$ with $L$; this can be considered the basepoint of $L$.) In the opposite direction, to each point $x$ of $L$, the line $L(p, x)$ intersects $C$ in $p$ and (since a quadratic with one root will also have another root) another point $q$ (which might be the same as $p$; this happens precisely when $L(p, x)$ is the tangent at $p$); this gives the inverse $\phi^{-1}(x) = q$. In this way we obtain an isomorphism $\phi: C \to L$ of subvarieties.

Working over an algebraically closed field $k$, where every nonsingular conic $C$ has a point, we may conclude that $C$ is isomorphic (as a projective variety) to $\mathbb{P}^1(k)$. Hence $C$ is a curve of genus $0$.

Projective duality

Working over an algebraically closed field $k$ (let us assume the characteristic is not $2$), all nondegenerate quadratic forms on a vector space $V$ are isomorphic and we may fix one as standard. For example, for $V = k^3$, we may fix attention on the quadratic form $q(x, y, z) = x^2 + y^2 + z^2$, which determines a conic section $C \subset \mathbb{P}^2(k)$ and an accompanying nondegenerate symmetric bilinear form $\langle-, -\rangle_q: V \times V \to k$.

Projective duality relative to $C$ is the projectivization of linear duality with respect to $\langle -, - \rangle$, which takes a linear subspace $L$ to its orthogonal dual $L^\perp = \{v \in V: (\forall w \in L)\; \langle v, w \rangle = 0\}$. We note that the orthogonal dual is an involution that takes joins of subspaces to meets and vice-versa. This construction descends through the quotient $k^3 \backslash \{0\} \to \mathbb{P}^2(k)$ to give an operation that takes points in $\mathbb{P}^2(k)$ (lines in $k^3$) to lines in $\mathbb{P}^2(k)$ (hyperplanes in $k^3$), and vice-versa, and moreover takes a join of two distinct points (the line incident to them) to the meet of their dual lines (the point of their intersection).

This duality may be visualized thus: given a nondegenerate conic $C$ and a point $p$ off of $C$, draw the two lines incident to $p$ that are tangent to $C$, and pass to the line incident to the tangent points. (This is easier to visualize by imagining $k = \mathbb{R}$ and considering a point $p$ exterior to say an ellipse $C$.) This defines the line that is projectively dual to $p$ (dual with respect to the conic $C$); if $p$ is on $C$, then the same procedure works by considering the two tangent points as infinitesimally close to $C$, so that the line between them is the tangent line at $p$: the projective dual of $p$ on $C$ is its tangent line (this is the case where the line in $k^3$ corresponding to $p$ is isotropic with respect to the bilinear form).

The entire procedure can be reversed and gives an anti-involution on the poset of flats of $\mathbb{P}^2(k)$, interchanging points and lines and interchanging meets and joins.

More generally, projective duality can be described in terms of an orthogonality map $\mathbb{P}(V) \to \mathbb{P}(V^\ast)$, where $V^\ast$ is the linear dual of $V$, which maps a subvariety to a corresponding dual “envelope” subvariety. This is explored in GKZ. The role of the conic section (or generally a conic hypersurface) is simply to set up an explicit self-duality $\mathbb{P}(V^\ast) \cong \mathbb{P}(V)$.

Pascal’s “mystic hexagon”

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• I.M. Gel’fand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser 2008 (paperback edition).

1. We took care to say pointed: note that depending on the field $k$, there might not even be a solution point on the conic $C$, even if $C$ is by all rights nonsingular. For example, consider $p(x, y) = x^2 + y^2 + 1$ over $\mathbb{R}$. Of course we can relax again if $k$ is algebraically closed.