Linear System of Divisors - Linear System of Conics

Linear System of Conics

For example, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L.

In the most elementary treatments a linear system appears in the form of equations

with λ and μ unknown scalars, not both zero. Here C and C′ are given conics. Abstractly we can say that this is a projective line in the space of all conics, on which we take

as homogeneous coordinates. Geometrically we notice that any point Q common to C and C′ is also on each of the conics of the linear system. According to Bézout's theorem C and C′ will intersect in four points (if counted correctly). Assuming these are in general position, i.e. four distinct intersections, we get another interpretation of the linear system as the conics passing through the four given points (note that the codimension four here matches the dimension, one, in the five-dimensional space of conics). Note that of these conics, exactly three are degenerate, each consisting of a pair of lines, corresponding to the ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient).