Divisor (Algebraic Geometry)¶
An algebraic-geometric divisor records codimension-one zero-and-pole data as a Weil cycle or compatible local Cartier equations, with class and line-bundle relations set by the space.
Core Idea¶
Divisors record codimension-one data on an algebraic space. A Weil divisor is a locally finite integer sum of prime codimension-one subschemes; a rational function gives a principal divisor from its orders of zeros and poles. Quotienting Weil divisors by principal ones gives Cl(X). Effective Cartier divisors instead require locally nonzerodivisor equations and produce invertible sheaves. The variants are related, not identical synonyms.[ref-175c45cf8a3f][ref-1d3316be2fc2][^ref-d7274138882e]
Scope of Application¶
On the smooth projective line P^1, closed points are prime divisors and the coordinate t gives div(t)=[0]−[∞]; thus the two point divisors are equivalent, not the same representative. On Spec(k[x,y]), the codimension-one axes V(x),V(y) give div(x/y)=[V(x)]−[V(y)]. k[x,y] is a UFD, so height-one primes are principal. These are explicit derivations from the Stacks definitions, not verbatim source examples.[ref-175c45cf8a3f][ref-74a730f64f20][ref-56fa3f705603][ref-f91215e9392f]
Clarity¶
The prime supports change with dimension: points on a curve, lines on an affine plane. Positive coefficient means a zero, negative a pole. A principal difference is zero in the class group even though its constituent loci remain distinct. On locally factorial spaces, Pic(X)→Cl(X) is an isomorphism; normality alone need not yield surjectivity.[ref-175c45cf8a3f][ref-06bde1e964e1]
Manages Complexity¶
Weil sums organize distributed zero/pole multiplicities; principal equivalence forgets coordinate-dependent representatives; Cartier equations connect geometry to line bundles. The distinctions matter at singularities, where codimension-one cycle data may persist without local invertibility.[ref-175c45cf8a3f][ref-06bde1e964e1][^ref-1cbf373ff4cd]
Abstract Reasoning¶
Specify the ambient integral scheme; list codimension-one supports; compute integer orders or check compatible local equations; then compare representatives modulo rational-function divisors. Before identifying Weil and Cartier classes, test local factoriality. An effective Cartier local equation must be a nonzerodivisor, not merely a single generator.[ref-175c45cf8a3f][ref-1d3316be2fc2][^ref-06bde1e964e1]
Knowledge Transfer¶
The zero/pole ledger transfers from projective curves to affine schemes, but “point = prime divisor” does not transfer to surfaces without a codimension check. Five redirect titles identify distinct variants, not aliases. This entry is unparented because a Weil-cycle parent would not cover all Cartier data without extra ambient-space hypotheses.[^ref-175c45cf8a3f]
[^ref-175c45cf8a3f]: Stacks Project, §31.27 Weil divisors. [^ref-1d3316be2fc2]: Stacks Project, §31.14 Effective Cartier divisors. [^ref-06bde1e964e1]: Stacks Project, Lemma 31.28.7. [^ref-d7274138882e]: Stacks Project, §71.7. [^ref-1cbf373ff4cd]: Stacks Project, §31.30. [^ref-74a730f64f20]: Stacks Project, Lemma 31.29.5. [^ref-56fa3f705603]: Stacks Project, Example 10.27.3. [^ref-f91215e9392f]: Stacks Project, Lemma 10.120.6.
Neighborhood in Abstraction Space¶
Divisor (Algebraic Geometry) sits in a sparse region of the domain-specific corpus (78th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Sheaves & Birational Geometry (22 abstractions)
Nearest neighbors
- Sheaf of Modules — 0.84
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Computed from structural-signature embeddings · 2026-10-08