We have seen that the circle and the sphere can each be defined by a single equation, namely and , respectively. However, the real projective line while the complex projective line , so can we think of the projective line as being defined by a single equation for any field ? And why do we call this a “line” anyway? In this lecture, we answer these questions and more.
Nonsingular Projective Varieties
Let’s begin with the precise language of projective varieties from algebraic geometry. We will be very technical in this first half of the lecture.
Let denote either , , or , even though the definitions hold for any number field. A subset in the form
is called a non-singular, irreducible, projective variety of dimension if the following axioms hold:
- Homogeneity: Each is homogeneous polynomial of degree , that is, for all nonzero .
- Nonsingularity: The matrix
has rank for each satisfying .
- Irreducibility: With being the set of homogeneous polynomials of degree , the set
is a prime ideal in the graded polynomial ring . That is, since we have the product , if and are homogeneous polynomials such that the product , then either or .
- Dimension: The integer is the number of homogeneous coordinates minus the number of hypersurfaces .
Here, I’ve given ad hoc definitions to match the situations we’ll focus on in this class. If you wish to see the most general framework, I’d suggest reading through Hartshorne’s Algebraic Geometry.
Affine vs. Projective Curves
In order to gain some intuition with these definitions, we begin with the simplest case: when is just defined by one equation. We continue to denote as either , , or .
The degree of the polynomial is the largest such that . An example of such a polynomial is where is an integer.
Make the substitution and so that we find the polynomial
We define the homogenization of to be the homogeneous polynomial of degree which occurs in the numerator above: . Note that by construction and . The collection of points such that is called a projective curve. As an example, if then its homogenization is
The notation in terms of the affine curve is shorthand for the collection of projective points:
Recall that we have an injective map which sends , so we may break the set above into two pieces:
The first set is essentially the collection of affine points, whereas the second set consists of those “points at infinity.”
We work through an example as a precursor to elliptic curves.
Consider the curve where . Then we have
We denote as the “point at infinity” on .
Proof: Let . (Note that this has degree .) Make the substitution and so that we find the homogeneous polynomial
Since we find that the collection of points at infinity consists of just the equivalence class .
Non-Singular Projective Curves
Say that we have a homogeneous polynomial of degree with coefficients in . We define a projective curve to be the collection of all projective points such that . We say is a non-singular projective curve if the gradient
does not vanish for any projective point . This is an example of a non-singular, projective variety of dimension . (Note that we have replaced by .) Any projective point such that is called a singular point. If is a nonsingular projective curve, its genus is the nonnegative integer .
Example: Non-Singular Projective Curve
We show a fundamental result about affine and projective lines.
Consider the curve . Then is a nonsingular projective curve of genus .
Proof: We check that is a nonsingular projective curve by using homogeneous coordinates. Upon substituting and , we find the expression so denote the polynomial . Its gradient is . Hence has no nonsingular points. Since has degree , we see that the genus of is .
Note that a line is a subset of . In fact, we have an embedding which sends ; so that we may identify as a line as well:
Since is a line, it has genus . We think of this line as the “line at infinity.”
In particular, if is a curve, then the set where is simply the intersection of the “line at infinity” with the projective curve . If is a polynomial of degree , then this set consists of only projective points (counting multiplicity).
Example: Singular Projective Curve
For the final example, we discuss what happens when the curve is singular. Following Cassels’s classic text Lectures on Elliptic Curves, let’s consider the cubic curve .
Upon denoting and , we find the projective curve in terms of the homogeneous polynomial
This is not a nonsingular projective curve. To see why, we compute the singular points. We have the partial derivatives
These simultaneously vanish when . Hence the projective point is the unique singular point in . You can also see from the graph above that the slope is not uniquely defined at the affine point .
In fact, staying away from the origin, we see that nonsingular points . We explain why. Define the map as that which sends . We’ll come back to this point later. Given any , the intersection with the curve and the line yields the point
(Actually, corresponds to so we get all of the points on , but the map fails to be one-to-one at this point.) In particular, the nonsingular points form a curve of genus — even though is a curve of degree . This means the genus formula we introduced before does not work when the projective curve is singular.
We will discuss more examples in the next lecture.