The genus of a smooth curve under every constant field extension #
Let X be a smooth, proper, geometrically integral curve over a field k, with function field
F = k(X). This file shows that the genus of F does not change under any algebraic extension
k' / k of the constants, separable or not:
g(F · k' / k') = g(F / k).
Over a perfect field, or for separable k' / k, this is Stichtenoth's Theorem 3.6.3(b)
(TauCeti.genus_eq_genus_of_constantCompositum_eq_top). Without separability the genus can drop
(TauCeti.GenusDrop.genus_lt_genus, the curve y² = x^p - t over 𝔽_p(t), whose regular proper
model is therefore not smooth), and in general it never increases
(TauCeti.genus_le_genus_of_constantCompositum_eq_top). Smoothness of the curve is what rules out
the drop, over any constant field.
The proof compares both genera with cohomology. The base change X_{k'} = X ×_k Spec k' is again
smooth, proper and geometrically integral, its genus dim H¹(X_{k'}, 𝒪) is that of X by flat
base change (TauCeti.AlgebraicGeometry.genus_eq_of_isPullback_of_smooth), and on each of these
curves the cohomological genus is the genus of the function field
(TauCeti.AlgebraicGeometry.SchemeWeilDivisor.genus_eq_genus_functionField). It remains to
identify the function field of X_{k'} with F · k', which holds for every geometrically integral
X (TauCeti.AlgebraicGeometry.nonempty_functionField_algEquiv_of_isPullback): both are the
fraction field of A ⊗[k] k' for a nonempty affine open U = Spec A of X.
For a finite extension k' / k the divisor theory of F carries over as well. Since F ⊗[k] k'
is a field, [F · k' : F] = [k' : k], and k' is the exact constant field of F · k'. With the
genus unchanged, no place of F becomes singular after extending the constants
(TauCeti.genus_eq_genus_iff_forall_constantSpanIntegers_eq). So L(Con D) = L(D) · k' and
ℓ(Con D) = ℓ(D) for every divisor D, the conorm carries the canonical class to the canonical
class, and it is injective on divisor classes. These are Stichtenoth's Theorem 3.6.3(d)–(f), which
he proves for perfect k; here k' / k may be inseparable.
Main results #
AlgebraicGeometry.Scheme.genus_eq_genus_functionField_of_smooth: the genusdim H¹(X, 𝒪_X)of a smooth proper geometrically integral curve is the genus of its function field.TauCeti.AlgebraicGeometry.genus_functionField_eq_of_isPullback: the function field of the base change of such a curve to any field extensionK / khas the same genus.TauCeti.AlgebraicGeometry.genus_eq_genus_functionField_of_constantCompositum_eq_top: the genus of the function field of a smooth proper geometrically integral curve is invariant under every algebraic constant field extension.TauCeti.AlgebraicGeometry.genus_eq_genus_of_constantCompositum_eq_top_of_algEquiv: the same for any fieldk-isomorphic to such a function field.TauCeti.AlgebraicGeometry.isEmpty_algEquiv_functionField_of_genus_lt: a field whose genus drops under some algebraic constant field extension is not the function field of a smooth proper geometrically integral curve.TauCeti.AlgebraicGeometry.riemannRochSpace_conorm_eq_span_of_smoothandTauCeti.AlgebraicGeometry.dim_conorm_of_smooth:L(Con D) = L(D) · k'andℓ(Con D) = ℓ(D)for a finite extensionk' / k.TauCeti.AlgebraicGeometry.conormClassGroup_canonicalClass_of_smoothandTauCeti.AlgebraicGeometry.conormClassGroup_injective_of_smooth: the conorm preserves the canonical class and is injective on divisor classes.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, second edition, Theorem 3.6.3.
- R. Hartshorne, Algebraic Geometry, GTM 52, Proposition III.9.3 (flat base change).
The genus of a smooth curve is the genus of its function field. For a smooth proper
geometrically integral curve X over k, the genus dim_k H¹(X, 𝒪_X) is the genus of the
function field k(X) defined through Riemann's theorem.
The function field of a smooth curve keeps its genus under base change. Let X be a
smooth proper geometrically integral curve over k and Y = X ×_k Spec K its base change to a
field extension K / k. Then the function field K(Y) has the genus of k(X).
The genus of a smooth curve is invariant under every constant field extension. Let X be
a smooth proper geometrically integral curve over k with function field F = k(X), and let
F' = F · k' for an algebraic extension k' / k, separable or not. Then g(F' / k') = g(F / k).
Without smoothness the genus can drop under an inseparable k' / k
(TauCeti.GenusDrop.genus_lt_genus).
The genus of a field with a smooth model is invariant under every constant field
extension. Let F / k be a field which is k-isomorphic to the function field of a smooth
proper geometrically integral curve X over k, and let F' = F · k' for an algebraic extension
k' / k, separable or not. Then g(F' / k') = g(F / k).
A field whose genus drops under a constant field extension has no smooth model. If
F' = F · k' for an algebraic extension k' / k and g(F' / k') < g(F / k), then F is not
k-isomorphic to the function field of any smooth proper geometrically integral curve over k.
Divisors of a smooth curve under finite constant field extensions #
Riemann–Roch spaces of a smooth curve under a finite constant field extension. Let X be a
smooth proper geometrically integral curve over k with function field F = k(X), and let
F' = F · k' for a finite extension k' / k, separable or not. Then for every divisor D of
F / k, L(Con D) is the k'-span of the image of L(D).
ℓ(Con D) = ℓ(D) on a smooth curve under a finite constant field extension, separable or
not: the Riemann–Roch dimensions of the function field of a smooth proper geometrically integral
curve over k do not change when the constants are extended to a finite k' / k.
The canonical class of a smooth curve under a finite constant field extension, separable
or not: the conorm sends the canonical class of the function field F = k(X) of a smooth proper
geometrically integral curve to the canonical class of F' = F · k'.
Divisor classes of a smooth curve under a finite constant field extension, separable or
not: the conorm on divisor classes of the function field of a smooth proper geometrically integral
curve over k is injective, so a divisor that becomes principal over k' is already principal.