The degree homomorphism on the Picard group, and Picβ° of a curve #
Let X be a proper integral curve over a field k whose codimension-one local rings are
discrete valuation rings and whose HΒΉ(X, πͺ_X) is finite-dimensional. Tensor product makes the
isomorphism classes of line bundles on X into the Picard group Pic X, and on it the
Euler-characteristic degree deg L = Ο(L) - Ο(πͺ_X) is additive. This file bundles that degree as
a homomorphism Pic X β+ β€ and defines Picβ° X, the degree-zero part of the Picard group, as
its kernel.
Picβ° X is then compared with the divisor-side degree-zero class group. Under the isomorphism
Cl(X) β
Pic X of TauCeti.AlgebraicGeometry.WeilDivisor.Scheme.Picard, the degree of a line
bundle is the residue-degree-weighted degree Ξ£_y D(y) [ΞΊ(y) : k] of any divisor D of it, so
the two degree-zero subgroups correspond. Composing with the description of the abstract Picβ°
as degree-zero divisors modulo principal divisors puts Picβ° X in the concrete form
Picβ° X β
{D : deg D = 0} / {div f}.
At a codimension-one point of residue degree one β for instance the image of a k-rational
point β the degree is surjective, so Pic X β§Έ Picβ° X β
β€.
Main declarations #
LineBundleClass.eulerDegreeHom, the degreePic X β+ β€, andLineBundleClass.picZero, the subgroupPicβ° Xit cuts out;SchemeWeilDivisor.eulerDegree_classGroupToLineBundleClass: the degree of the line bundle of a divisor class is the weighted degree of that class;SchemeWeilDivisor.classGroupPicZeroAddEquivPicZero, the isomorphismClβ°(X) β Picβ°(X), andSchemeWeilDivisor.weightedDegreeZeroQuotientAddEquivPicZero, which presentsPicβ° Xas the degree-zero divisors modulo the principal ones;SchemeWeilDivisor.eulerDegreeHom_surjectiveandSchemeWeilDivisor.picQuotientPicZeroAddEquivInt: at a point of residue degree one the degree is ontoβ€, soPic X β§Έ Picβ° X β β€.
References #
- R. Hartshorne, Algebraic Geometry, Chapter II, Section 6 and Chapter IV, Section 1.
- Q. Liu, Algebraic Geometry and Arithmetic Curves, Chapter 7, Section 3.
The degree homomorphism deg : Pic X β+ β€ of a proper curve over k, sending the class of a
line bundle L to Ο(L) - Ο(πͺ_X). The Picard group is written additively, as Additive of the
tensor-product monoid of line-bundle classes.
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The degree homomorphism evaluates to the Euler-characteristic degree.
Picβ° X, the degree-zero part of the Picard group of a proper curve over k: the kernel
of the Euler-characteristic degree.
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Picβ° X is the kernel of the degree.
A line-bundle class lies in Picβ° X exactly when its degree vanishes.
The degree on the Picard group computes the weighted degree of a divisor class. Under
Cl(X) β
Pic X, the Euler-characteristic degree of the line bundle of a divisor class is the
residue-degree-weighted degree Ξ£_y D(y) [ΞΊ(y) : k] of that class.
Cl(X) β
Pic X carries the degree-zero divisor classes onto Picβ° X.
Clβ°(X) β
Picβ°(X). On a proper integral curve over k whose codimension-one local rings
are discrete valuation rings, D β¦ πͺ_X(D) identifies the degree-zero divisor classes with the
degree-zero part of the Picard group.
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Clβ°(X) β
Picβ°(X) is the restriction of Cl(X) β
Pic X.
The inverse of Clβ°(X) β
Picβ°(X) is the restriction of the inverse of Cl(X) β
Pic X.
Picβ° X is the group of degree-zero divisors modulo principal divisors.
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Divβ°(X) / (principal divisors) β
Picβ° X sends the class of a degree-zero divisor D to the
class of πͺ_X(D).
The degree is onto β€ at a point of residue degree one. A codimension-one point with
residue field k β for instance the image of a k-rational point β carries a line bundle of
every degree.
Pic X β§Έ Picβ° X β
β€ on a proper curve carrying a codimension-one point of residue
degree one, the isomorphism being the Euler-characteristic degree.
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The isomorphism Pic X β§Έ Picβ° X β
β€ sends the class of a line bundle to its degree.