Summary: Agebraic Geomerty I
Lectures 6 and 7
October 20, 2008
1 Projective Space
In Euclidean geometry, two distinct lines in the plane intersect in exactly in
one point except if they are parallel; this exception is a bit of a nuisance.
So we add one extra point to the Euclidean plane for each direction (these
are called points at infinity) and declare that parallel lines meet at the point
at infinity corresponding to their common direction. The corresponding
geometry is called projective geometry. Just like Euclidean geometry has
an axiomatic definition, so does projective geometry. The former has an
analytic description in terms of the "usual" coordinate geometry (Descarte).
An analytic description of projective geomerty was introduced by Pl¨ucker,
Mobius, and others.
1.1 General Idea
Put the xy-plane in 3-dimensions and shift it up by 1. Each point on the
xy-plane is in correspondence with a line in through the originwhich is not
in the xy-plane. Each line in the xy-plane corresponds to a direction. So
we can think of the projective plane as the set of lines through the origin;