Curves of genus zero and the projective line #
Let X be a regular proper curve over a field k, that is, a curve over k whose local rings at
its codimension-one points are discrete valuation rings, and suppose that k is integrally closed
in the function field k(X). If X has genus zero, dim_k H¹(X, 𝒪_X) = 0, and carries a divisor
of degree one, for instance the prime divisor of a k-rational point, then X is the projective
line: there is an isomorphism X ≅ ℙ¹_k over k. For a k-rational point the hypothesis on k
is automatic: a rational point forces k to be integrally closed in k(X)
(AlgebraicGeometry.Scheme.isIntegrallyClosedIn_functionField_of_section).
This is the geometric form of Stichtenoth's Proposition 1.6.3. The genus of X is the genus of
k(X) (TauCeti.AlgebraicGeometry.SchemeWeilDivisor.genus_eq_genus_functionField), and a divisor
of degree one on X is a divisor of degree one of k(X), so the function field is rational
(TauCeti.nonempty_algEquiv_ratFunc_of_genus_eq_zero_of_divisor_degree_eq_one). A k-isomorphism
k(X) ≃ k(t) then identifies X with ℙ¹_k through the two-chart models of the two function
fields: X is the two-chart model of k(X) (AlgebraicGeometry.Scheme.isoTwoChartModel), an
isomorphism of function fields induces an isomorphism of two-chart models
(TauCeti.AlgebraicGeometry.twoChartModel.isoOfAlgEquiv), and ℙ¹_k is the two-chart model of
k(t) (TauCeti.AlgebraicGeometry.projectiveLine.isoTwoChartModel).
Genus zero alone does not suffice: the function field of a conic without rational points has genus
zero but no divisor of degree one
(TauCeti.not_exists_divisor_degree_eq_one_of_sq_add_sq_add_one_eq_zero).
It does suffice after a separable extension of degree at most two, for smooth curves. Let X be a
smooth proper geometrically integral curve of genus zero over k. Its function field has exact
constant field k and a separating element, since the differentials Ω_{k(X)/k} are the generic
stalk of the invertible sheaf Ω_{X/k}, so it has a place of degree at most two whose residue field
is separable over k (TauCeti.exists_place_degree_le_two_and_isSeparable_of_genus_eq_zero). That
place is the place of a closed point x of X, with the same residue field. Over any extension
K of k into which κ(x) embeds, the base change X_K is again smooth, proper, geometrically
integral and of genus zero, and it has a K-rational point, so it is ℙ¹_K. Taking K = κ(x),
X becomes the projective line over a separable extension of degree at most two.
Main definitions #
AlgebraicGeometry.Scheme.isoProjectiveLine: the isomorphismX ≅ ℙ¹_koverkinduced by ak-isomorphismk(t) ≃ k(X).
Main results #
AlgebraicGeometry.Scheme.functionFieldMap_isoProjectiveLine_hom: the pullback of rational functions alongX.isoProjectiveLine eise.AlgebraicGeometry.Scheme.nonempty_algEquiv_ratFunc_of_genus_eq_zero_of_divisor_degree_eq_one: a regular proper curve of genus zero with a divisor of degree one has rational function field.AlgebraicGeometry.Scheme.exists_iso_projectiveLine_of_genus_eq_zero_of_divisor_degree_eq_one: such a curve is isomorphic toℙ¹_koverk.AlgebraicGeometry.Scheme.exists_iso_projectiveLine_of_genus_eq_zero_of_section: a regular proper curve of genus zero with ak-rational point is isomorphic toℙ¹_k, with no hypothesis on the constants.AlgebraicGeometry.Scheme.exists_iso_projectiveLine_of_isPullback_of_genus_eq_zero: a smooth proper geometrically integral curve of genus zero becomesℙ¹_Kover every extensionKin which it has a point.AlgebraicGeometry.Scheme.exists_finrank_residueField_le_two_and_isSeparable_of_genus_eq_zero: such a curve has a point whose residue field is separable of degree at most two overk.AlgebraicGeometry.Scheme.exists_finrank_le_two_and_iso_projectiveLine_of_genus_eq_zero: such a curve becomesℙ¹_Kover a separable extensionK / kof degree at most two.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 1.6.3 and Remark 1.6.4.
- The Stacks Project, Chapter 55, Semistable Reduction, the case of genus zero.
- R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977, Corollary I.6.12 and Example IV.1.3.5.
The isomorphism induced by a rational function field #
A curve with rational function field is the projective line: the isomorphism X ≅ ℙ¹_k
induced by a k-isomorphism e : k(t) ≃ k(X), through the two-chart models of k(X), built from
the function e t, and of k(t), built from t. Its pullback of rational functions is e
(functionFieldMap_isoProjectiveLine_hom).
Instances For
The isomorphism X ⟶ ℙ¹_k of isoProjectiveLine is a morphism over k.
The pullback of rational functions along X ≅ ℙ¹_k is e: under the identification of the
function field of ℙ¹_k with k(t), the pullback along X.isoProjectiveLine e is e.
Curves of genus zero #
A curve of genus zero with a divisor of degree one has rational function field. Let X be
a regular proper curve over k, with k integrally closed in k(X). If dim_k H¹(X, 𝒪_X) = 0
and some divisor on X has degree one, then k(X) is k-isomorphic to the rational function
field k(t).
A curve of genus zero with a divisor of degree one is the projective line. Let X be a
regular proper curve over k, with k integrally closed in k(X). If dim_k H¹(X, 𝒪_X) = 0 and
some divisor on X has degree one, for instance the prime divisor of a k-rational point, then
X is isomorphic to ℙ¹_k over k.
A curve of genus zero with a rational point is the projective line. Let X be a regular
proper curve over k. If dim_k H¹(X, 𝒪_X) = 0 and X has a k-rational point, then X is
isomorphic to ℙ¹_k over k. No hypothesis on the constants is needed: the rational point forces
k to be integrally closed in k(X) (Scheme.isIntegrallyClosedIn_functionField_of_section).
Smooth curves of genus zero over a separable extension of degree at most two #
A smooth curve of genus zero is the projective line over every field in which it has a
point. Let X be a smooth proper geometrically integral curve over k with
dim_k H¹(X, 𝒪_X) = 0, let K / k be a field extension and Y = X ×_k K. If some point x of
X has residue field κ(x) embedding into K over k, that is, if X has a K-valued point,
then Y is isomorphic to ℙ¹_K over K.
A smooth curve of genus zero has a point of degree at most two with separable residue
field. Let X be a smooth proper geometrically integral curve over k with
dim_k H¹(X, 𝒪_X) = 0. Then some point x of X has residue field κ(x) a separable extension of
k of degree at most two.
A smooth curve of genus zero becomes the projective line over a separable extension of
degree at most two. Let X be a smooth proper geometrically integral curve over k with
dim_k H¹(X, 𝒪_X) = 0. Then there is a separable extension K / k of degree at most two such that
X ×_k K is isomorphic to ℙ¹_K over K.