On the other hand, it’s a compact Riemann surface

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Prelimanary

1. An Example

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+Zλ₂with C = 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 ho- momorphism (algebra), holomorphic function (complex analysis), and covering (topology). A function ¯f on E gives a function f on C which is double periodic, i.e.,

f (z + λ₁) = f (z), f (z + λ₂) = f (z), ∀z ∈ C.

Recall that we have the following well-known result:

Theorem 1.1 (Liouville). A bounded entire function, i.e. a holo- morphic function on C with bounded image, is constant.

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 doubly periodic holomorphic function f on C such that f ◦π = f . It then suffices to claim that a a doubly periodic holomorphic¯ function on C is bounded.

To do this, consider D := {z = α₁λ₁ + α₂λ₂|0 ≤ α_i ≤ 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 func- tion on E with only a simple pole

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Hint. If the pole is not on the boundary of D, then consider the path integral along the boundary of D, otherwise consider the path integral along a translation of D. By using Residue theorem, one can see it.

Then the next hope 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).

And P⁰ is a functions with pole of order 3. By direct computation, one sees that

{wp_equation}

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.

Exercise 1.5. Verify that P is doubly peiodic, i.e. P(z+λ) = P(z) for all z ∈ C and λ ∈ Λ.

Work out the computation in Lemma 1.4.

Theorem 1.6. An elliptic curve can be embedded into P²_C as a non- singular cubic.

Sketch. We considering the map ϕ : E − {0} → C² given by

¯

z 7→ (P(z), P⁰(z)).

This map can be extended to ϕ : E → P² as

½ ϕ(¯z) = [P(z), P⁰(z), 1] if ¯z 6= 0, ϕ(¯z) = [0, 1, 0] if ¯z = 0.

The affine defining equation in C² is y² = 4x³− g2x − g3. And the projective defining equation is

y²z = 4x³− g2xz²− g3z³.

One can verify that this cubic is non-singular and the map ϕ is an

embedding. ¤

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2. Divisors

Similar phenomena occurs for any compact Riemann surface. As the above section suggested, the essential point is to find enough functions and then determine the algebraic relation between those functions. All these can be done for any compact Riemann surface. Thus the purpose of this section is to show the following

Theorem 2.1. 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 di- visors.

Definition 2.2. Let X be a compact Riemann surface. A divisor, denoted D = P

n_iP_i, is a finite formal sum of finite points (codim=1).

Given divisors D1 =P

niPi and D2 =P

miPi, one can define D₁+ D₂ :=X

(n_i+ m_i)P_i.

Let Div(X) be the set of all divisors on X. It’s clear that Div(X) is a free abelian group under the addition defined above. In fact, one can think Div(X) as the free abelian group on the set X.

Given a meromorphic function f on X, one can count its zeros and poles with multiplicity. This give rise to the following group homomor- phism

div : M(X) − {0} → Div(X),

where M(X) denotes the field of meromorphic functions on X.

Exercise 2.3. A meromorphic function on a compact Riemann surface X has at most finitely many zeros and poles.

Hence div is well-defined.

The idea for divisors is to collect information on poles and zeros.

We denote the functions with prescribed poles and zeros, collected in D =P

n_iP_i, as

L(D) := {f ∈ M(X) − {0}|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 n_i.

Example 2.4. Let X = C ∪ {∞} be the Riemann sphere. Then M(X) ∼= C(z). We use the notation of [x] to denote the divisor of the point with coordinate z = x.

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

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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 ∞.

We recall some definitions and properties of divisors.

Definition 2.5. Let D = P

n_iP_i be a divisor. We said that D is effective if n_i ≥ 0 for all i, denoted D ≥ 0.

WE write D₁ ≥ D₂ if D₁− D₂ ≥ 0.

Definition 2.6. Given divisors D, D⁰, they are said to be linearly equivalent, denoted D ∼ D⁰, if D − D⁰ = div(f ) for some f ∈ M(X).

The linear series |D| is thus defined to be the set

|D| := {D⁰ ∈ Div(X)|D⁰ ≥ 0, D⁰ ∼ D}.

Notice that we have a induced natural map π : L(D) − {0} → |D|,

f 7→ div(f ) + D.

If one identify L(D) ∼= Cⁿ, then |D| can be identified as Pⁿ⁻¹. In particular, one has that if L(D) 6= 0, then

dim |D| = dim L(D) − 1.

It is interesting an important to determine the dimension of L(D), denoted l(D). We recall the following fact.

{p1} Proposition 2.7. Let D = P

n_iP_i be a divisor. We said that D is effective if n_i ≥ 0 for all i, denoted D ≥ 0. Suppose now that D is an effective non-zero divisor then L(−D) = {0}.

We leave the proof as an exercise. (Hint: prove that a holomorphic function on a compact Riemann surface must be constant).

Among all divisors, there is a most important one, called canonical divisor, denoted K_X. It is a divisor associate to meromorphic 1-form.

Example 2.8. Let X = C ∪ {∞} be the Riemann sphere. X can be covered by coordinate charts (U₀, z), (U₁, w), where U₀ = X − {∞} ∼= C and U1 = X − {0} ∼= C. Note that w = ¹_z.

We consider a holomorphic 1-form dz on U0. This can be extended into a meromorphic 1-form on U1, because near ∞, z = _w¹ and hence dz = ⁻¹_w2dw. Therefore, the extended meromorphic 1-form gives rise to a divisor −2[∞]. This is a canonical divisor of X.

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Warning. One can pick other meromorphic 1-form and then pro- duce a different canonical divisor. However, all these are linearly equivalent. (Exercise: check it). Therefore, to be more precise, one should consider the equivalent classes of divisors of 1-forms. Nevertheless, in most real application, the quantity that we are going to compute is the same inside the equivalent class. Therefore, we usually abuse the notation by picking any one inside the class and call it the canonical divisor K_X.

In general, l(D) can be computed or estimated by the most impor- tant theorem for curves, the Riemann-Roch theorem:

Theorem 2.9 (Riemann-Roch).

l(D) − l(K_X − D) = deg(D) + 1 − g(X),

where K_X denotes the canonical divisor, l(D) denotes the dimension of L(D), and g(X) is the genus of the curve X.

We will leave the sketch of the proof to next section. In this section, we concentrate on its applications.

We now redo the example of elliptic curve with the help of Riemann- Roch theorem.

Let E = C/Λ be an elliptic curve. Then the genus is 1. The 1-from dz on C is doubly periodic, hence induced a 1-form dz on E. Thus K_X = 0.

Let D 0 be an effective divisor. By Prop. 2.7, K = 0, and Riemann-Roch, we have:

l(D) = deg(D).

Let ¯0 be the image of π(0) in E.The vector space L(k[¯0]) has a nat- ural basis {1}, {1, P}, {1, P, P⁰} respectively when k = 1, 2, 3. How- ever, when k = 6, one finds that l(6[¯0]) = 6 and thus {1, P, P⁰, P², PP⁰, P³, P⁰²} must be linearly dependent. Hence there must be a relation between them involving P³, P⁰².

By using L(3[¯0]), one produces a meromorphic map ϕ : E 99K P(L(3[¯0])) = P².

The linear dependency shows that the image satisfies a cubic polyno- mial. This recover the embedding given in the previous section.

Turning to a compact Riemann surface X in general. Following the above construction, we would like to ask if there is a divisor D such that L(D) has enough sections in the sense that

ϕ_D : X 99K P(L(D)), is an embedding.

By Riemnann-Roch Theorem, one can prove that:

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Theorem 2.10. Let X be a compact Riemann surface of genus g.

Suppose that D is a divisor with deg(D) ≥ 2g + 1, then ϕD is en embedding.

Sketch. ¤

3. Riemann-Roch Theorem 4. Algebraic Varieties