Conics
Let K be a field. Let P be the projective plane over K as introduced here. Let's temporarily define a conic C in P as a subset of points defined by an equation of the form
C : F = 0
where
F = a_{00} X_0^2 + a_{01} X_0X_1 + a_{02} X_0X_2 + a_{11} X_1^2 + a_{12} X_1X_2 + a_{22} X_2^2
with a_{ij} ∈ K. What does this mean? Well, if (a : b : c) is a point of P then we can substitute X_0 = a, X_1 = b, X_2 = c into F. If we get zero we say the point is on the conic and if not we say the point is not on the conic. Note that this makes sense, because if (a : b : c) = (a' : b' : c') in P, then there exists a nonzero scalar λ in K such that a = λ a', b = λ b', c = λ c' and we see that
F(a, b, c) = λ^2 F(a', b', c')
because F is homogeneous of degree 2 in X_0, X_1, X_2. Thus whether or not we get zero depends only on the point of P and not on the particular choice of 3-vector representing it.
But there is something really wrong with saying that a conic is a set of points, as you can see when you do the following
Exercise 10: Find a field K and a conic as defined above without any points.
Instead we simply say that a conic in P is given by a homogeneous degree 2 polynomial F in X_0, X_1, X_2. Moreover, we say F and F' define the same conic if and only if F = λ F' for some nonzero scalar λ ∈ K. We always use the phrase “Let C : F = 0 be a conic in P” to indicate this situation.
Exercise 11: Show that every conic is either a line (counted double), a union of two lines, or has the property that it meets every line in 0, 1, or 2 points.
Exercise 12: Write a script that tells you which of the three cases happens (here K is Z/pZ).
If the equation F of a conic is reducible, then we say the conic is reducible. Otherwise we say the conic is irreducible. If the equation F remains irreducible when replacing the field K by its separable algebraic closure we say that C is geometrically irreducible.
Exercise 13: Show that given a geometrically irreducible conic C : F = 0 the partial derivatives F_0, F_1, F_2 of F by X_0, X_1, X_2 have no common zero, except possibly if the characteristic of K is 2. Can you understand what goes wrong in characteristic 2?
Exercise (optional) 14: Show that every conic over Z/pZ has a point. (This exercise is optional because the goal of the reu is to prove/find something geometric and this exercise is essentially arithmetic in nature.)
Suppose that C is an irreducible conic and that P is a point of C. Then we can parametrize C as is explained beautifully in the slides by Damiano Testa (which you can find on the page with reading materials. For the moment we mean by this a way of “parametrizing” the points of C by a single parameter t.
Exercise 15: Write a script that takes as input a conic C over Z/pZ and either spits out “reducible” or finds a parametrization of C.
What is a parametrization really? A bit more precise would be that we want a morphism P^1 —> P from the projective line to P whose image is the curve C.