Logarithmic pair¶
In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities.
Core Idea¶
Logarithmic pair is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. A boundary Q-divisor on a variety is a Q-divisor D of the form Σd i D i where the D i are the distinct irreducible components of D and all coefficients are rational numbers with.
Scope of Application¶
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Documented setting. In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities.
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Definition. A boundary Q-divisor on a variety is a Q-divisor D of the form Σd i D i where the D i are the distinct irreducible components of D and all coefficients.
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Definition. A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D.
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Definition. The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X.
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Definition. A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form.
Clarity¶
A clear use of Logarithmic pair names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities.
Manages Complexity¶
Logarithmic pair compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—d log(z) = dz/z along components of the divisor given locally by z=0.—and the practical consequence—a logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form.
Abstract Reasoning¶
- Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities.
- Check operation and conditions. A boundary Q-divisor on a variety is a Q-divisor D of the form Σd i D i where the D i are the distinct irreducible components of D and all coefficients are rational numbers with 0≤d i.
Knowledge Transfer¶
Within the home domain. Knowledge about Logarithmic pair transfers literally when a new case preserves the same carrier type, relation, and recognition test. In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. A boundary Q-divisor on a variety is a Q-divisor D of the form Σd i D i where the D i are the distinct irreducible components of D and all coefficients are rational numbers with 0≤d i ≤1. Beyond the home domain. No canonical parent is asserted for Logarithmic pair.
Relationships to Other Abstractions¶
Current abstraction Logarithmic pair Domain-specific
Parents (1) — more general patterns this builds on
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Logarithmic pair is part of Algebraic Variety Domain-specific
An algebraic variety is the ambient constituent inside the larger logarithmic pair, together with its distinguished divisor.
Hierarchy path (1) — routes to 1 parentless root
- Logarithmic pair → Algebraic Variety
Neighborhood in Abstraction Space¶
Logarithmic pair sits in a sparse region of the domain-specific corpus (72nd 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
- Mordellic Variety — 0.85
- Terminal singularity — 0.84
- Zero Divisor — 0.83
- 2–3 Heap — 0.83
- Dimension of an algebraic variety — 0.83
Computed from structural-signature embeddings · 2026-10-08