nLab divisor (algebraic geometry)

Idea

In algebraic geometry a divisor (or Weil divisor for definiteness) in a given variety is a formal linear combination of sub-varieties of codimension 1. Accordingly the divisor group is the free abelian group on the set of subvarieties of codimension 1.

A divisor is called effective if all its coefficients are positive numbers.

In complex analytic geometry any meromorphic function induces a divisor whose summands are irreducible codimension-1 subvarieties weighted by the order of the poles of the function on that subvariety. Divisors arising this way are called principal divisors. More generally in algebraic geometry this leads to the concept of Cartier divisor.

Two divisors are called linearly equivalent if their difference is a principal divisor.

Properties

Relation to line bundles and the Picard group

Under suitable conditions the Poincaré dual of a divisor may hence define the first Chern class of an algebraic line bundle and this construction yields group homomorphism from the group of divisors to the Picard group. After quotienting out linear equivalence of divisors this yields an inclusion and in complex geometry this inclusion exhibits precisely the subgroup of the Picard group of holomorphic line bundle which admit at least one global holomorphic section (e.g. Huybrechts 04, cor. 2.3.20).

More abstractly in terms of abelian sheaf cohomology:

A divisor $D = \sum_i n_i D_i$ on $X$ induces a short exact sequence of abelian sheaves on $X$ (Brylinski 94, (1-1)) of the form

$$0 \to \mathcal{O}^\times_X \longrightarrow \mathcal{O}_X(\ast Y)^\times \stackrel{v_Y}{\longrightarrow} (v_Y)_\ast \mathbb{Z}_{\tilde Y} \to 0 \,,$$

where $Y$ is the support of $D$ (i.e. $Y \coloneqq \underset{i | n_i \neq 0}{\cup} D_i$) and $\nu_Y \colon \tilde Y \to Y$ is its normalization in $Y$ (…). The induced long exact sequence in abelian sheaf cohomology has as first connecting homomorphism a map of the form

$$\delta \colon H^0(\tilde Y, \mathbb{Z}) \longrightarrow H^1(X,\mathcal{O}^\times_X) \,.$$

Now the divisor $D = \sum_i n_i D_i$ is incarnated as the locally constant function $n$ on $\tilde Y$ whose value on $\tilde D_i$ is $n_i \in \mathbb{Z}$, hence the divisor is incarnated a class in $H^0(\tilde Y, \mathbb{Z})$. The class $\mathcal{O}(D)$ of the line bundle it induces as the image of this class under the above connecting homomorphims:

$$\mathcal{O}(D) \simeq \delta(n) \,.$$

(Brylinski 94, prop. 1.1).

A higher analog of the connecting homomorphism above may be interpreted as being the Beilinson regulator from $K_1(X)$ to Deligne cohomology in degree 3 (Brylinski 94, section 3).

Examples

• canonical divisor

• theta divisor

Related concepts

• algebraic cycle

• exceptional divisor (resolution of singularities)

References

• Wikipedia, Divisor (algebraic geometry)

• The Stacks Project, 53 Divisors on Algebraic Spaces

In complex analytic geometry:

• Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, Astérisque 226 (1994): 145-174 (pdf)

• Daniel Huybrechts section 2.3 of Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)

Relation of divisors to complex cobordism cohomology:

• Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467-493 (doi:10.1090/S0894-0347-97-00232-4)

• MO discussion