Plane conics

Moduli
The moduli space of plane conics is a 5-dimensional projective space of homogeneous degree two polynomials in 3 variables, up to scaling.

Dual conics
The dual curve of a smooth plane conic is a smooth conic in the dual P2. The condition that the dual conic pass through a point in the dual P2 is equivalent to the condition that the original conic be tangent to the corresponding line, and vice versa.

Enumerative geometry

 * Through any five points, no three of which are collinear, there is a unique smooth conic. (This is found by intersecting hyperplanes in the moduli space.)
 * There is a unique conic tangent to five fixed general lines. (This is proved using the above statement and dual conics.
 * There are 3264 conics tangent to five fixed general conics.

Special cases

 * Line pairs in the plane
 * Double lines in the plane

Generalizations

 * Quadric hypersurfaces
 * Plane curves
 * Degree two curves in projective space