Castelnuovo's inequality #
Let F / k be an algebraic function field with exact constant field k, and let F₁, F₂ be
subfields of F containing k, of genera g₁, g₂, over which F is finite of degrees
n₁ = [F : F₁] and n₂ = [F : F₂]. Castelnuovo's inequality bounds the genus of F:
g ≤ n₁ g₁ + n₂ g₂ + (n₁ - 1) (n₂ - 1).
This file proves it first whenever F₁ has a place P₁ that splits completely in F into places
whose restrictions to F₂ are pairwise distinct and rational, and F₂ has at least g₂ further
rational places. When F has a separating element, with F = F₁ F₂, the inequality then holds
unconditionally; this covers every function field over a perfect constant field, and the
separably generated ones over an imperfect constant field. At least one of F / F₁ and F / F₂
is separable: otherwise the universal k-differential vanishes on both subfields, hence on F,
contradicting dim_F Ω[F⁄k] = 1.
After interchanging the subfields if necessary, write a primitive element y of F / F₁ as
∑ cᵢ zᵢ with cᵢ ∈ F₁ and zᵢ ∈ F₂, and take a place P₁ of F₁ at which the minimal
polynomial of y has separable reduction, the cᵢ are regular, the zᵢ are regular at every
place above, and F / F₁ is unramified at every place above. A separating element supplies
infinitely many places of F with residue field separable over k, so P₁ can be taken among
their restrictions to F₁; then every place above P₁ has residue field separable over k, and
so do g₂ + n₁ places of F₂, chosen the same way. Pass to a finite Galois constant field
extension F · k' into which all these residue fields embed, over which F · k', F₁ · k' and
F₂ · k' have the genera of F, F₁ and F₂, and [F · k' : Fᵢ · k'] = [F : Fᵢ]. There every
place above P₁ is rational and F₂ · k' has enough rational places. Kummer's theorem then
gives n₁ places over P₁ at which y takes distinct constant values. The presentation of y
computes each of these values from the restrictions of the place to F₁ and to F₂, so the
places have distinct restrictions to F₂. This finite separable extension replaces Stichtenoth's
passage to an algebraic closure of k, which over an imperfect field could lower the genus.
The proof chooses an F₁-basis of F made of functions of F₂ with few poles, and feeds it to
the bound g ≤ 1 + n₁ (g₁ - 1) + deg C for a basis of F / F₁ inside L(C)
(TauCeti.genus_le_one_add_finrank_mul_genus_sub_one_add_degree). Let P⁽⁰⁾, …, P⁽ⁿ¹⁻¹⁾ be the
places of F over P₁ and Q⁽ⁱ⁾ their restrictions to F₂. Take a nonspecial effective
divisor B of F₂ of degree g₂ away from the Q⁽ⁱ⁾ (Stichtenoth, Proposition 1.6.12). For
i ≠ 0, Riemann's theorem gives ℓ(B + Q⁽ⁱ⁾) ≥ 2 > ℓ(B), so some wᵢ ∈ L(B + Q⁽ⁱ⁾) has a pole
at Q⁽ⁱ⁾ and none at the other Q⁽ʲ⁾; put w₀ = 1. In a relation ∑ aᵢ wᵢ = 0 with
aᵢ ∈ F₁, the coefficients have the same order at every P⁽ʲ⁾, namely e (P⁽ʲ⁾ ∣ P₁) times
their order at P₁; looking at the place P⁽ⁱ⁾ of a coefficient of least order shows that the
i-th term has strictly least order there, which is impossible. So the wᵢ are a basis of
F / F₁ inside L(A), A = B + ∑_{i ≠ 0} Q⁽ⁱ⁾, of degree g₂ + n₁ - 1, and the conorm of A
to F has degree n₂ (g₂ + n₁ - 1).
The split-place form of the inequality does not assume F = F₁ F₂: the basis above consists of
elements of F₂. The hypothesis enters the general theorem only through the presentation of y.
Main results #
TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add_of_isSplitCompletely: Castelnuovo's inequalityg ≤ n₁ g₁ + n₂ g₂ + (n₁ - 1) (n₂ - 1), from a place ofF₁splitting completely inFwith distinct rational restrictions toF₂.TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add: Castelnuovo's inequality forF = F₁ F₂with a separating element, over any constant field, without a separability hypothesis on either subextension.TauCeti.genus_le_finrank_mul_genus_add_finrank_mul_genus_add_of_perfectField: the same over a perfect constant field, where a separating element always exists.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 3.11.1 and Theorem 3.11.3.
Castelnuovo's inequality (Stichtenoth, Theorem 3.11.3), from a split place. Let F / k
have exact constants and let F₁, F₂ be subfields of F containing k over which F is
finite. Suppose a place P₁ of F₁ splits completely in F, the places of F over it restrict
to pairwise distinct rational places of F₂, and F₂ has a set T of at least g(F₂) rational
places containing none of these restrictions. Then
g(F) ≤ [F : F₁] g(F₁) + [F : F₂] g(F₂) + ([F : F₁] - 1) ([F : F₂] - 1).
Neither k nor the subfields need be perfect or separable over anything; only the constants of
F have to be exact.
Castelnuovo's inequality (Stichtenoth, Theorem 3.11.3). Let F / k be a function field
with exact constants and a separating element x, and let F₁, F₂ be subfields of F
containing k with F = F₁ F₂ and F finite over both. Then
g(F) ≤ [F : F₁] g(F₁) + [F : F₂] g(F₂) + ([F : F₁] - 1) ([F : F₂] - 1).
The constant field k may be imperfect, and neither subextension is assumed separable.
Castelnuovo's inequality over a perfect constant field (Stichtenoth, Theorem 3.11.3): if
F = F₁ F₂ is finite over both subfields and has exact constants, then
g(F) ≤ [F : F₁] g(F₁) + [F : F₂] g(F₂) + ([F : F₁] - 1) ([F : F₂] - 1).
Over a perfect field every function field has a separating element.