COMPLEX GEOMETRY Course notes

Theorem 2.8.2. Let P (T ) = T n + c 1 T n−1 + · · · + c n ∈ M(Z)[T ] be an irreducible polynomial in T over Z.
Then there exists a finite map of Riemann surfaces f : Z ′ −→ Z of degree n, unique up to isomorphisms,
and a meromorphic function F on Z ′ satisfying
F n + (f ∗ c 1 )F n−1 + · · · + (f ∗ c n ) = 0
Corollary 2.8.1.
(1) Z 1 −→ Z 2 finite =⇒ f ∗ : M(Z 2 ) ϕ
↩→ M(Z 1 ) is a finite field extension of degree n.
(2) Conversely, any finite field extension of degree n gives rise to a finite map Z 1 −→ Z 2 whose associated
field extension is isomorphic to ϕ.
(3) A field extension K of transcendence degree 1 (i.e., K is a finite field extension over C(Z)) is isomorphic
to M(Z) for some Riemann surface Z (finite over CP 1 ).
Z is called a smooth model of K.
Theorem 2.8.3. Let C ⊆ CP 2 be an irreducible plane curve, i.e., C = V (F ) where F is an irreducible
homogeneous polynomial in (z 0 : z 1 : z 2 ). Then there exists a compact Riemann surface Z and a generically
injective map f : Z −→ CP 2 whose image is C.
Here, generically injective means birational (←→ isomorphic on an open set).
Proof: If F is irreducible then C[x 0 , x 1 , x 2 ]/(F, x 2 − 1) is an integral domain and has a quotient field
K. The mapping f(P ) = (x 0 (P ) : x 1 (P ) : 1) extends to by continuity to a desingularization of C.
Theorem 2.8.4 (Hurwitz). If f : Z 1 −→ Z 2 is a finite map, then K Z1 ∼ f ∗ K Z2 + R (f ∗ = pullback) where
R is the ramification divisor ∑ p∈Z 1
r p (f) · p, r p (f) = ord p (f) − 1, where f : Z 1 −→ Z 2 is locally at p of the
form f(z) = z ordp(f) and
deg(f) =
∑
ord p (f),
p∈ fibre
where this sum is independent of the choice of the fibre.
Corollary 2.8.2 (Riemann-Hurwitz).
deg(K Z1 ) = (deg(f)) · (deg(K Z2 )) + deg(R)
Corollary 2.8.3. g(Z 1 ) ≥ g(Z 2 ), where g is for genus.
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