The Riemann–Roch theorem for a proper curve #
Let X be a proper integral curve over a field k whose codimension-one local rings are
discrete valuation rings, with a k-rational point and with H¹(X, 𝒪_X) finite-dimensional.
For every Weil divisor D on X,
χ(𝒪_X(D)) = dim_k H⁰(X, 𝒪_X(D)) - dim_k H¹(X, 𝒪_X(D)) = deg D + 1 - g,
where deg D = Σ_y D(y) [κ(y) : k] is SchemeWeilDivisor.relativeDegree (X ↘ Spec k) and
g = dim_k H¹(X, 𝒪_X) is the genus. This is the combination of χ(𝒪_X(D)) = deg D + χ(𝒪_X)
with χ(𝒪_X) = 1 - g, the latter being where the rational point enters: it forces the global
functions to be the constants.
Two consequences are recorded. The first is Riemann's inequality deg D + 1 - g ≤ dim_k H⁰(𝒪_X(D)),
obtained by discarding H¹. The second is that a divisor of negative degree has no nonzero global
sections, so that its H¹ has dimension exactly g - 1 - deg D: a nonzero global section of
𝒪_X(D) is a rational function f with div f + D ≥ 0, and the degree of that effective divisor
is deg D, because degree is a linear-equivalence invariant.
Riemann's inequality also makes the Riemann–Roch space of a divisor of degree at least the genus
nonzero, so such a divisor is linearly equivalent to an effective divisor: nonemptiness of a
complete linear system is the existence of a nonzero global section, by
SchemeWeilDivisor.nonempty_completeLinearSystem_iff_nontrivial_globalSections_sheaf.
Main declarations #
SchemeWeilDivisor.eulerCharBelow_sheaf_eq_relativeDegree_add_one_sub_genusandSchemeWeilDivisor.finrank_cohomology_zero_sheaf_sub_finrank_cohomology_one_sheaf: the Riemann–Roch theorem, for the Euler characteristic and in terms of the two dimensions;InvertibleSheaf.eulerCharBelow_eq_relativeDegree_add_one_sub_genus: Riemann–Roch for a line bundle presented as𝒪_X(D);SchemeWeilDivisor.relativeDegree_add_one_sub_genus_le_finrank_cohomology_zero_sheaf: Riemann's inequality;SchemeWeilDivisor.nonempty_completeLinearSystem_of_genus_le_relativeDegree: a divisor of degree at least the genus is linearly equivalent to an effective divisor;SchemeWeilDivisor.sections_top_eq_bot_of_relativeDegree_neg,SchemeWeilDivisor.finrank_cohomology_zero_sheaf_eq_zero_of_relativeDegree_negandSchemeWeilDivisor.finrank_cohomology_one_sheaf_eq_of_relativeDegree_neg: a divisor of negative degree has no global sections, and the resulting value ofdim H¹.
References #
- R. Hartshorne, Algebraic Geometry, Chapter IV, Theorem 1.3 and Corollary 1.3.2.
- Q. Liu, Algebraic Geometry and Arithmetic Curves, Chapter 7, Theorem 3.17.
A divisor of negative degree has no nonzero global sections. On a proper integral curve
over a field k whose codimension-one local rings are discrete valuation rings, with
H¹(X, 𝒪_X) finite-dimensional, the Riemann–Roch space Γ(X, 𝒪_X(D)) of a divisor of negative
degree is zero.
The Riemann–Roch theorem. On a proper integral curve over a field k whose
codimension-one local rings are discrete valuation rings, with a k-rational point and with
H¹(X, 𝒪_X) finite-dimensional, every Weil divisor D satisfies
χ(𝒪_X(D)) = deg D + 1 - g,
where χ(M) = dim H⁰(X, M) - dim H¹(X, M), deg D = Σ_y D(y) [κ(y) : k] and g is the
genus.
The Riemann–Roch theorem, in terms of the two cohomology dimensions. On a proper integral
curve over a field k whose codimension-one local rings are discrete valuation rings, with a
k-rational point and with H¹(X, 𝒪_X) finite-dimensional,
dim_k H⁰(X, 𝒪_X(D)) - dim_k H¹(X, 𝒪_X(D)) = deg D + 1 - g.
Riemann's inequality. On a proper integral curve over a field k whose codimension-one
local rings are discrete valuation rings, with a k-rational point and with H¹(X, 𝒪_X)
finite-dimensional, dim_k H⁰(X, 𝒪_X(D)) ≥ deg D + 1 - g.
On a proper integral curve over a field k whose codimension-one local rings are discrete
valuation rings, with H¹(X, 𝒪_X) finite-dimensional, H⁰(X, 𝒪_X(D)) vanishes for a divisor
D of negative degree.
On a proper integral curve over a field k whose codimension-one local rings are discrete
valuation rings, with a k-rational point and with H¹(X, 𝒪_X) finite-dimensional, a divisor of
negative degree has dim_k H¹(X, 𝒪_X(D)) = g - 1 - deg D.
A divisor of degree at least the genus is linearly equivalent to an effective divisor.
Riemann's inequality makes the Riemann–Roch space of such a divisor nonzero, and a nonzero global
section of 𝒪_X(D) names an effective divisor in the class of D. Properness over the field
makes X Noetherian, which supplies the order system used by the complete linear system.
The Riemann–Roch theorem for a line bundle. On a proper integral curve over a field k
whose codimension-one local rings are discrete valuation rings, with a k-rational point and with
H¹(X, 𝒪_X) finite-dimensional, a line bundle L ≅ 𝒪_X(D) satisfies
χ(L) = deg D + 1 - g.