Algebraic surfaces
Sep. 17, 2003 (Wed.)
1. Comapct Riemann surface and complex algebraic curves This section is devoted to illustrate the connection between compact Riemann surface and complex algebraic curve. We will basically work out the example of elliptic curves and leave the general discussion for interested reader.
Let Λ⊂C be a lattice, that is, Λ = Zλ1+Zλ2 withC = Rλ1+Rλ2. Then an elliptic curve E :=C/Λ is an abelian group.
On the other hand, it’s a compact Riemann surface. One would like to ask whether there are holomorphic function or meromorphic functions on E or not.
To this end, let π :C → E be the natural map, which is a group homomorphism (algebra), holomorphic function (complex analysis), and covering (topology). A function ¯f on E gives a function f onC which is double periodic, i.e.,
f (z + λ1) = f (z), f (z + λ2) = f (z), ∀z ∈ C.
Recall that we have the following
Theorem 1.1 (Liouville). There is no non-constant bounded entire function.
Corollary 1.2. There is no non-constant holomorphic function on E.
Proof. First note that if ¯f is a holomorphic function on E, then it induces a double periodic holomorphic function on C. It then suffices to claim that a a double periodic holomorphic function on C is bounded.
To do this, consider D := {z = α1λ1+ α2λ2|0 ≤ αi ≤ 1}. The image of f is f (D). However, D is compact, so is f (D) and hence f (D) is bounded. By Liouville
theorem, f is constant and hence so is ¯f .
The next hope is to ask if there is a meromorphic function on E with only a simple pole or not. The answer is NO, which can be proved by residue theorem.
Exercise 1.3. Prove that there is no non-zero meromorphic function on E with only a simple pole
Then the next step is to look for functions with pole of order 2. Luckily, we have one, which is the Weierstrass P-function,
P(z) := z−2+ X
ω∈Λ−{0}
((z− ω)−2− ω−2).
AndP0 is a functions with pole of order 3. By direct computation, one sees that Lemma 1.4.
P0(z)2= 4P(z)3− g2P(z) − g3. Where
g2= 60 X
ω∈Λ−{0}
ω−4
and
g3= 140 X
ω∈Λ−{0}
ω−6.
By considering the map E→C2(or toP2) given by ¯z 7→ (P(z), P0(z)), then one realize the elliptic curve as a cubic curve in P2. The affine defining equation inC2 is
y2= 4x3− g2x − g3. And the projective defining equation is
y2z = 4x3− g2xz2− g3z3.
1
2
We therefore realize an elliptic curve as an algebraic curve.
Similar phenomena occurs for any compact Riemann surface. As the above example suggested, the essential point is find enough functions and then determine the algebraic relation between those functions. All these can be done for any compact Riemann surface. Thus we have the following
Theorem 1.5. Any compact Riemann surface can be embedded into a projective space as an algebraic curve.
To study functions more systematically, it’s natural to consider divisors.
Definition 1.6. Let X be a compact Riemann surface/ non-singular algebraic curve, a divisor, denoted D =P
niPi, is a finite formal sum of points (codim=1).
The idea for divisors is to collect information on poles and zeros. We denote the functions with prescribed poles and zero as
L(D) := {f ∈ M(X)|div(f) + D ≥ 0}.
It’s clearly a vector space over the ground field and its dimension is denoted l(D).
Another important notion for divisor is the degree, which is deg(D) :=X
ni.
Let’s look at the definition a little bit more. Given a meromorphic function (or rational function) f (z), one can collect all the poles and zeros together with orders at these points, give rise to a divisor div(f ).
Example 1.7. Let X = C ∪ {∞} be the Riemann sphere. Let f1(z) = z, f2(z) = 1/(z − 1), f3(z) = z/(z − 1)2. By easy computation, one finds that div(f1) = 1[0]−1[∞] since it has a zero at 0 and a pole at ∞. Similarly, div(f2) = 1[∞]−1[1], div(f3) = 1[0] + 1[∞] − 2[1].
Now fix a divisor D = 2[1]. What is L(D)? What does it mean? It is nothing but the set of meromorphic functions with at most a pole of order 2 at [1] and no other poles. More precisely,
L(D) = {g(z)/(z − 1)2|deg(g(z)) ≤ 2}.
Because if deg(g(z)) ≥ 3 then it gives a pole at ∞.
Now we can state the most important theorem for curves, the Riemann-Roch theorem:
Theorem 1.8 (Riemann-Roch).
l(D) − l(K − D) = deg(D) + 1 − g(X),
where K denotes the canonical divisor, and g(X) is the genus of the curve X.
We will turn to the discussion of canonical divisor more thoroughly later. At this moment, it’s enough to know that it’s a divisor of degree 2g − 2 and for an elliptic curve, K = 0.
Before we move on, we recall an easy fact:
Proposition 1.9. 1 Let D =P
niPi be a divisor. We said that D is effective if ni ≥ 0 for all i, denoted D ≥ 0. Suppose now that D is an effective non-zero divisor then L(D) = {0}.
Proof. Exercise. (Hint: prove that a holomorphic function on a compact Riemann
surface must be constant)
We now redo the example of elliptic curve with the help of Riemann-Roch the- orem. It’s a fact that the following argument also works for any compact Riemann surface.
3
Example 1.10. Let E =C/Λ be an elliptic curve. Then the genus is 1. Let ¯0 be the image of π(0) in E. By Prop. 1.91 and K = 0, Riemann-Roch reads:
l(D) = deg(D).
The vector space L(k¯0) has a natural basis {1, P}, {1, P, P0} respectively when k = 2, 3. However, when k = 6, one finds that {1, P, P0, P2,PP0,P3, P02} must be linearly depedent. Hence there must be a relation between them involving P3, P02.
On the other hand, for a complex algebraic curve X, it is non-singular if and only if one can find a local chart at each point. Hence a non-singular complex algebraic curve is a Riemann surface. If X is projective, then it’s a closed subset in a compact setPN. Therefore it is compact.
We’ve seen that an elliptic curve E can be embedded intoP2 as a non-singular cubic curve. There is a group structure on E. How this group structure behave on cubic curves? In fact there is a nice geometric correspondence via intersection.
