Varieties are a basic structure in algebraic geometry. They were the central objects of study before Grothendieck reinvented the entire theory in his treatise Éléments de géométrie algébrique by introducing schemes. In the following we will introduce varieties and define algebraic curves.
We will restrict our discussion to varieties over the real or complex numbers. As such, let be or .
We begin by discussing varieties over affine space. Let be affine -space over (this is just as a set but we’re “forgetting” the vector space structure). We will denote points of by . Let denote the polnomial ring in variables over . If is an ideal, we call the subset
of an affine algebraic set. By construction is the common zero-set of . In other words, is the largest set for which all functions vanish on. It is best to illustrate this with a few examples.
- Affine -space is an algebraic set if is the zero ideal.
- Let be the ideal generated by . Since elements of are multiples of , any , satisfies or equivalently . Thus is the unit circle.
- Let be the ideal generated by . Similar to the previous example, any satisfies or . Therefore is the union of the line and the parabola .
The takeaway from these examples is that algebraic sets, geometrically, are curves (and in higher dimensions surfaces, hypersurfaces, etc.). We say that an affine algebric set is an affine algebraic variety if the ideal corresponding to is prime. In the above examples, the first two are algebraic varieties while the third is not. Notice that the algebraic set in the last example turned out to be a union of two graphs. This holds more generally; affine algebraic varieties are irreducible in the sense that they cannot be written as nontrivial unions of smaller algebraic sets.
In general, affine space is not “nice enough” to work with. For example, elliptic curves are certian varieties with a group structure and affine space does not allow for a natural choice of identity element in the group. To get around this, we will work with varities over projective space which resolves this issue by introducing specified points “at infinity”. In the case of elliptic curves, there is a single point “at infity” which will be a natural choice for the identity element.
We define projective -space over , denoted by , to be the set of all equivalence classes of -tuples with at least one where if there exists a such that for all . We denote the class of by and points by . Since any line in affine -space is of the form
and all of these points are identified in (in fact this is exactly the class ), one can think of as the set of lines in . We call the set
the points at infinity of . To see why this is the case, notice that the map
identifies and . Moreover the map
identifies and . So, can also be thought of as a copy of and the points at infinity (or ). Moreover, we could of required any to get different copies of and inisde .
- Notice always consists of a single point. Thus is a plane (the complex plane) with a point at infinity while is a line (the real line) with a point at infinity.
Index the variables of by . We say that a polynomial is homogenous of degree if
for all . We say an ideal is homogenous if it is generated by homogenous polynomials. Let denote the subset of consisting of homogenous polynomials. If , it makes sense to ask if since this is independent of the representative of . To each homogenous ideal , we call the subset
of a projective algebraic set. Similar to the affine case, is the common zero-set of .
- Projective -space is a projective algebraic set if is the zero ideal.
- A line in is the algebraic set corresponding to a homogenous ideal .
A projective algebraic variety is a projective algebraic set where the ideal is prime. Similar to the affine case, projective algebraic varieties are irreducible in the sense that they cannot be written as nontrivial unions of smaller projective algebraic sets.
Coordinate Rings and Dimension
To give both affine and projective varities the notion of dimension, we need to introduce coordinate rings. Apart from letting us define the dimension of a variety coordinate rings also play a critial role in the general theory.
The defintion is the same in both the affine and projective setting. So, let be a variety (either affine or projetive) with prime ideal . The coordinate ring of is the quotient ring defined by
This defintion should make sense because the elements (which we also call polynomials) of are defined on up to a polynomial which vanishes on . In other words, an element of is an equivalence class of polynomials where elements of a class agree when their domain is restricted to . Since is a field, is an integral domain and moreover is an integral domain because is prime. Therefore, we define the function field of to be the field of fractions of .
If is a commutative ring, then we say that a chain of prime ideals of the form
has length (so the length is the number of strict inclusions). We define the Krull dimension of to be the supremum of chains of prime ideals of . In particular, the Krull dimension need not be finite. If is an affine variety, then the dimension of , denoted is defined to be the Krull dimension of (this ring is commutative since is commutative and quotients respect commutativity). The Krull dimension is, in general, not easy to calculate. But we do have a nice theorem:
Theorem: If is an affine variety, then the dimension of is equal to the trancendence degree of over .
In particular, this tells is that the dimension of is finite. This also makes computing the dimension of an affine variety significantly easier.
- The dimension of is because so that , and this clearly has trancendence degree over .
- If the ideal corresponding to an affine variety is generated by a single nonconstant polynomial equation, then the dimension of the variety is . This is because relates any of the by a polynomial (hence algebraic) relation in the other variables.
So why didn’t we just define the dimension of an affine variety to be the trancendence degree of over ? It’s because the Krull dimension definition is intuitive. Standard ring theory tells us that prime ideals of are in bijective correspondence with prime ideals of containing . If is prime then as varieties (we are being verbose here by writing for ). In this case we say is an affine subvariety of (there is an obvious analgous defintion for projective subvariety). So, if
is maximal chain (we can say maximal since the dimension of is finite) of prime ideals in , then there is a correponding chain of varieties
This correspondence is bijective because the previous correspondence is. So geometrically, the only way to make a variety smaller is by reducing its dimension (because then the corresponding chain of prime ideals decreases in length). To define the notion of dimension for projective varities we need a proposition.
Proposition: Let be a projective variety. Then for any there exists a copy of inside such that and is an affine variety.
With this proposition, we define the dimension of a projective variety , denoted , to be the dimension of as an affine variety. This is well-defined as it can be shown that the dimension of is independent of the copy of . The idea behind the definition of dimension for projective varieties is that the points at infinity shouldn’t be accounted for because they are a dimension lower than the affine part (recall can be viewed as together with ).
The Defintion of Algebriac Curves
An algebraic curve (or simply a curve) is just a projective variety of dimension one. If is a homogenous polynomial and is the projective variety with corresponding ideal , then it’s not too hard to show is an algebraic curve. To compute the dimension, choose a copy of inside . Then show generates the ideal corresponding to and use it to prove is of transcendence degree one over .
The Arithmetic of Elliptic Curves – Joseph Silverman