##  [Divisor](/divisor-0) 

 Definition

A formal finite Z-linear combination of codimension-one irreducible subvarieties (prime divisors) on an algebraic variety, used to record the orders of zeros and poles of rational functions or of sections of line bundles.

 

 

 

 

 

 





## Principle

Principle

Divisors encode local vanishing orders along codimension-one loci and organize those data up to linear equivalence; principal divisors come from logarithmic orders of rational functions and Cartier divisors correspond to local single-valued equations where defined.

 

 

 

 

 





## Demonstration

Demonstration

On a smooth projective curve, a nonzero rational function f defines the principal divisor (f)=Σ_v ord_v(f)·v, summing zeros with positive multiplicity and poles with negative multiplicity; on a surface a curve component appears as a codimension-one term with an integer coefficient.

 

 

 

 

## Misapplication

Misapplication

Treating any subvariety as a divisor regardless of codimension, or assuming every Weil divisor is Cartier on a nonnormal variety; or identifying a divisor with a single global equation where none exists.

 

 

 

 

 





## Consequence

Consequence

Correct use yields divisor class groups and Picard groups, a bridge to line bundles and linear systems, and intersection numbers that control geometry and enumerative invariants.

 

 

 

 

## Reversal

Reversal

Negating coefficients turns effective divisors into anti-effective ones (zeros become poles); forgetting linear equivalence replaces divisor classes by raw cycles, losing information about associated line bundles.

 

 

 

 

 





## Boundary

Boundary

Applies only to codimension-one cycles; distinctions matter between Weil and Cartier divisors, principal divisors, effective divisors, and divisor classes — some equivalences require normality or smoothness.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Divisor competes with related notions such as general cycles, subvarieties of other codimension, and line bundles/invertible sheaves; the tension is between cycle-theoretic raw data and equivalence classes that reflect geometric line bundles.

 

 

 

 

 





## Synthesis

Synthesis

A divisor is the codimension-one cycle that records where and with what multiplicity sections or functions vanish or blow up; modulo principal divisors it classifies the twisting captured by associated line bundles.