Seventh day of class - 9/15
Announcement: Office hours are normally held from 4 pm to 5 pm today. It will have to
happen a little bit later than usual.
Last time - Finished up our discussion of group theory that we will need. We also started
talking about projective space.
Projective space
Recall
Defn: Projective space P2 is the set of triples [a, b, c], where a, b and c are not all zero, and
where we say that [a, b, c] ∼ [a0 , b0 , c0 ] if there is a nonzero number t so that a0 = at, b0 = bt and
c0 = ct.
Any point in P2 has the form [a, b, 1] (and the set of these points are the “ordinary” points in
the plane), or [a, b, 0]. This is called a “point at infinity”, and it corresponds to an asymptotic
direction that the curve goes.
Ex: Let C be the curve given by
f (x, y) = x2 + xy − 2y 2 + x − 5y + 7 = 0.
Let F (X, Y, Z) be the homogenization of f and C̃ : F (X, Y, Z) = 0 be the projective closure
of C in P2 . What are the points at infinity on C̃?
What’s good about projective space?
Answer 1 - It is “compact.” (For example, you can also think of P2 as the set of all lines
through the origin in P2 . You can say that the distance between two lines is the (smallest)
angle between them, and this means that any two points in P2 have distance at most π/2 from
each other!)
Answer 2 - Things that are usually true in affine space are *always* true in projective space.
Theorem: Any two (different) lines in P2 intersect in exactly one point.
A line in P2 is defined by an equation ax + by + cz = 0.
This generalizes.
Theorem: Any line intersects a conic (degree 2 curve) in exactly two points (counting multiplicity).
We could draw a tangent line to a conic.
Defn: The “intersection index” or “multiplicity” of a point P on two curves C1 and C2 is a
non-negative integer I(C1 ∩ C2 , P ) reflecting the extent to which C1 and C2 are tangent to one
another at P , or one or the other is not smooth at P .
1
2
Theorem: Bezout’s theorem states that
X
I(C1 ∩ C2 , P ) = (deg C1 )(deg C2 ).
P ∈C1 (C)∩C2 (C)
Examples:
1. x + y + 2 = 0 and x2 + y 2 = 1.
2. x + 1 = 0 and x2 − y = 0.
3. x + y = 2 and x2 + y 2 = 2.
Q: How do you feel about projective space? Are there more examples I can do or talk about
that will make it feel a bit more comfortable?
Note: If C is a curve in projective space and (a : b : c) is a point on C, we can choose a, b and
c to be integers that are relatively prime. Then for any prime p, we can reduce a, b and c mod
p. The map (a : b : c) 7→ (a mod p : b mod p : c mod p) is a map from C(Q) to C(Fp ). This is
quite handy, since there isn’t actually a map from Q to Fp .