The Euler-characteristic degree of an effective Cartier divisor #
Let X be a scheme over a field k and I ⊆ 𝒪_X an ideal sheaf whose closed subscheme
D = V(I) is finite over k, of rank dim_k Γ(D, 𝒪_D). The exact sequence
0 ⟶ I ⟶ 𝒪_X ⟶ ι_* 𝒪_D ⟶ 0 and the vanishing of higher cohomology on the affine Noetherian
scheme D give
χ(I) = χ(𝒪_X) - dim_k Γ(D, 𝒪_D)
(Scheme.IdealSheafData.eulerCharBelow_sheaf, in
TauCeti.AlgebraicGeometry.IdealSheaf.EulerCharacteristic), where χ is the degree-2 truncation
dim H⁰ - dim H¹ and H⁰(X, 𝒪_X), H¹(X, 𝒪_X) are assumed finite-dimensional. When I is the
ideal sheaf 𝒪_X(-D) of an effective Cartier divisor D, this says that the line bundle 𝒪_X(-D)
has Euler-characteristic degree -dim_k Γ(D, 𝒪_D)
(Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_toInvertibleSheaf).
Twisting the exact sequence by a line bundle L with finite-dimensional H⁰ and H¹ gives
χ(L ⊗ I) = χ(L) - dim_k Γ(D, 𝒪_D) (the rank-one case of
Scheme.IdealSheafData.eulerCharBelow_tensor_sheaf), so
twisting by 𝒪_X(-D) and by 𝒪_X(D) is additive on degrees
(Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_mk_mul_mk_toInvertibleSheaf and
Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_mk_mul_inv_mk_toInvertibleSheaf), without
any integrality or regularity of X. In particular 𝒪_X(D) has degree dim_k Γ(D, 𝒪_D)
(Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_inv_mk_toInvertibleSheaf); on a connected
smooth proper curve over k the finiteness hypotheses hold automatically
(Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_inv_mk_toInvertibleSheaf_of_smooth).
This is the classical deg 𝒪_X(D) = deg D for effective divisors, with the degree of D read off
as the rank of its closed subscheme, the notion of degree used for relative effective Cartier
divisors in families.
The rank is Mathlib's Scheme.Hom.finrank of the finite morphism D ⟶ Spec k, evaluated at the
unique point of Spec k; it is dim_k Γ(D, 𝒪_D) by
AlgebraicGeometry.Scheme.Hom.finrank_eq_finrank_of_field.
References #
- R. Hartshorne, Algebraic Geometry: Proposition II.6.18 (
𝒪_X(-D)is the ideal sheaf ofD), Lemma III.2.10 and Theorem III.3.5 (the cohomology of𝒪_D), and the proof of Theorem IV.1.3 (Riemann–Roch), where degrees are computed from the same kind of exact sequence. - The Stacks Project, Section 33.44: degrees on proper curves, and the change of Euler characteristic under twisting by an effective Cartier divisor.
The degree of 𝒪_X(-D). On a scheme over a field k with finite-dimensional
H⁰(X, 𝒪_X) and H¹(X, 𝒪_X), the ideal sheaf 𝒪_X(-D) of an effective Cartier divisor D
finite over k has Euler-characteristic degree -dim_k Γ(D, 𝒪_D), the negative of the rank of
D ⟶ Spec k.
Twisting by 𝒪_X(-D) lowers the degree by deg D. On a scheme over a field k with
finite-dimensional H⁰(X, 𝒪_X) and H¹(X, 𝒪_X), let D be an effective Cartier divisor finite
over k and L a line bundle with finite-dimensional H⁰(X, L) and H¹(X, L). Then
deg (L ⊗ 𝒪_X(-D)) = deg L + deg 𝒪_X(-D). No integrality or regularity of X is needed.
Twisting by 𝒪_X(D) raises the degree by deg D. On a scheme over a field k with
finite-dimensional H⁰(X, 𝒪_X) and H¹(X, 𝒪_X), let D be an effective Cartier divisor finite
over k and L a line bundle with finite-dimensional H⁰(X, L) and H¹(X, L). Then
deg (L ⊗ 𝒪_X(D)) = deg L - deg 𝒪_X(-D), the class of 𝒪_X(D) being the inverse of that of
𝒪_X(-D).
The degree of 𝒪_X(D). On a scheme over a field k with finite-dimensional
H⁰(X, 𝒪_X) and H¹(X, 𝒪_X), as for a proper curve, the line bundle 𝒪_X(D) of an
effective Cartier divisor D finite over k has Euler-characteristic degree dim_k Γ(D, 𝒪_D),
the rank of the finite morphism D ⟶ Spec k. No integrality or regularity of X is needed.
The degree of 𝒪_X(D) on a smooth curve. On a connected smooth proper curve over a
field k, the line bundle 𝒪_X(D) of an effective Cartier divisor D has Euler-characteristic
degree dim_k Γ(D, 𝒪_D), the rank of the finite morphism D ⟶ Spec k. The geometric
hypotheses supply integrality, discrete valuation rings in codimension one and the finiteness of
H¹(X, 𝒪_X); no rational point or perfect-field assumption is needed.