Projective Conics

Abstract

The general equation of a conic will be written

$$q = a{x^2} + a'{y^2} + a''{z^2} + 2byz + 2b'zx + 2b''xy$$

In all of this chapter, P = P(E) is a projective plane over a commutative field K of characteristic ≠ 2; we put P* = P(E*). We will often identify a point m ∈ P with its homogeneous coordinates (x,y, z).

We will generally fix a conic α ∈ PQ(E) and its image C = im(α) (in most cases, α will be proper and have non-empty image), as well as one equation q for α. For a a point of C, the tangent to C at a will sometimes be denoted by 〈a, a〉).