The genus of the two-chart model

Let F / k be an algebraic function field with exact constant field k, and let X_F = TauCeti.AlgebraicGeometry.twoChartModel hF hx be its regular proper model, glued from the normalizations of k[x] and k[x⁻¹] in F for some x ∈ F transcendental over k. This file compares the cohomology of X_F with the Riemann–Roch theory of F.

The cohomological genus dim_k H¹(X_F, 𝒪_{X_F}) is the genus of F, defined through Riemann's theorem. This is SchemeWeilDivisor.genus_eq_genus_functionField for the curve X_F, transported along the identification k(X_F) ≃ₐ[k] F of TauCeti.AlgebraicGeometry.twoChartModel.functionFieldEquiv.

When F has genus one, the ladder ℓ(n P) = n deg P of Riemann–Roch spaces (TauCeti.Divisor.dim_natCast_zsmul_ofPoint_of_genus_eq_one) becomes the ladder dim_k H⁰(X_F, 𝒪_{X_F}(n P)) = n deg P for n ≥ 1, where the place P is regarded as a point of X_F through TauCeti.AlgebraicGeometry.twoChartModel.equivPlace. At a rational place this is h⁰(X_F, 𝒪_{X_F}(n P)) = n.

Main results

All in the namespace TauCeti.AlgebraicGeometry:

• twoChartModel.genus_eq_genus: the genus dim_k H¹(X_F, 𝒪_{X_F}) of X_F is g(F).
• twoChartModel.finrank_cohomology_zero_sheaf_natCast_zsmul_ofPoint_of_genus_eq_one: in genus one, dim_k H⁰(X_F, 𝒪_{X_F}(n P)) = n deg P for every place P of F and n ≥ 1.

References

• H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Theorem 1.5.17 and Appendix B.
• R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977, Section IV.1.

theorem TauCeti.AlgebraicGeometry.twoChartModel.genus_eq_genus {k F : Type u} [Field k] [Field F] [Algebra k F] {hF : IsFunctionField k F} {x : F} {hx : x ≠ 0} (hxt : Transcendental k x) (hk : IsIntegrallyClosedIn k F) : AlgebraicGeometry.Scheme.genus k (twoChartModel hF hx) = genus k F

The genus of the two-chart model is the genus of the function field. For x transcendental over k and k the exact constant field of F, the genus dim_k H¹(X_F, 𝒪_{X_F}) of X_F is g(F).

theorem TauCeti.AlgebraicGeometry.twoChartModel.finrank_cohomology_zero_sheaf_natCast_zsmul_ofPoint_of_genus_eq_one {k F : Type u} [Field k] [Field F] [Algebra k F] {hF : IsFunctionField k F} {x : F} {hx : x ≠ 0} (hxt : Transcendental k x) (hk : IsIntegrallyClosedIn k F) (hg : genus k F = 1) (P : Place k F) {n : ℕ} (hn : 1 ≤ n) : Module.finrank k (Scheme.Modules.Cohomology (SchemeWeilDivisor.sheaf (↑n • WeilDivisor.ofPoint ((equivPlace hxt).symm P))) 0) = n * P.degree

The section-dimension ladder on the two-chart model of a genus-one function field. For x transcendental over k, k the exact constant field of F, and F of genus one, every place P of F and every n ≥ 1 satisfy dim_k H⁰(X_F, 𝒪_{X_F}(n P)) = n deg P, where P is regarded as a point of X_F. At a rational place this reads dim_k H⁰(X_F, 𝒪_{X_F}(n P)) = n.