In the previous lecture, we introduced the notion of a non-singular projective curve. Today we discuss examples in detail by focusing on Elliptic Curves, Cubic Curves, and Quartic Curves.
As always, we will denote as either , , or . We begin with a standard definition.
Consider the proejctive curve , where the .
- Using the substitution
we find a cubic curve in the form , where
- is a nonsingular curve if and only if .
A nonsingular curve as in the proposition above is called an elliptic curve. Note that the discriminant of the cubic is , so the cubic curve is a nonsingular curve if and only if the cubic has distinct roots.
Proof: Following the ideas from the previous lecture, consider the homogeneous cubic polynomial
Its gradient involves the partial derivatives
It suffices to consider either the “affine” points or the point at infinity . For the latter, we have , showing that is never a singular point. For the former, we set and then use the substitutions above:
We seek all and such that these three partial derivatives vanish. Clearly we must have , , and . It is easy to see that these equations have a solution if and only if . .
Let me mention in passing that is the unique singular point on if and only if — assuming that not both and are zero, of course.
We discuss a few examples of cubic equations and whether they are singular or nonsingular.
- Consider the curve . We have and . Then , so is indeed an elliptic curve.
- Now consider the curve . We have and . Then , so is also an elliptic curve.
- Finally, consider the curve . We must place it in the proper form, so substitute and . We find that , so we set and . In this case, , so is not an elliptic curve.
We present another say to see is not an elliptic curve by explicitly finding a singular point. The homogenization of the polynomial is the cubic polynomial which has the gradient . If is a singular point, we must have the four equations
It is easy to check that the only solution is , so is indeed a singular point. Since there is a singular point on is it not a nonsingular projective curve.
There are really only four types of graphs for curves in the form .
- If , then there is one “egg”. Consider the graph of .
- If there are two. Consider the graph of .
- If yet , then we have a “node.” Consider the graph of .
- If because when we have a “cusp.” Consider the graph of .
Here are some general observations about cubic polynomials. The most general polynomial of degree is
(Note that there are 10 monomials in this expression.) The homogenization is the polynomial
The “points at infinity” on the curve correspond to the roots of the equation . If is nonsingular, then it has genus .
Now that we’ve seen cubic curves, let’s focus on quartic curves. We’ll show in some cases that these quartic curves are actually elliptic curves.
Consider the curve . If (1) there exists a -rational point on and (2) the quartic has distinct roots in , then may also be expressed in the form , where
such that . Moreover, the substitution between the two curves can be chosen to have coefficients in .
Proof: The discriminant of the quartic is nonzero because has distinct roots. We break the proof into two cases for the point .
Case #1: . Denote the polynomials
Make the substitutions
in terms of the constants
It is straight-forward to check that with and as above. In fact, , so that is a -rational point on the cubic curve.
Case #2: . In this case, the quartic has a distinct -rational root $x_0$. Say that we may factor in the form , where the are complex roots. Without loss of generality, assume that is the -rational root, and note the identities
Since each of these expressions is also in . We make the substitution
Note that these transformations have coefficients in . .