Terminal singularity¶
In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Core Idea¶
Terminal singularity is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. In mathematics, canonical singularities are a class of singularities that appear on the canonical model of an algebraic variety, and terminal singularities are a narrower class that occur as singularities of minimal models. These classes of singularities were introduced by Miles .
Scope of Application¶
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Pairs. The definitions in this paragraph are the ones used when a log resolution is not known to exist, for example when the base field has positive characteristic and X has dimension.
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Explanation. This suggests that pairs can be viewed as geometric objects comparable to varieties: for some purposes, X can be replaced by the pair (Y,\Delta) .
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Definition. Let X be a normal variety over a field whose canonical class K X is \mathbb Q -Cartier (as discussed below), and let f\colon Y \to X be a resolution.
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Definition. where the sum is over the exceptional divisors of f (the codimension-1 subvarieties of Y, these being irreducible by definition, whose image in X has codimension at least 2).
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Then X is said to be. (One can also say that X has "terminal singularities" or "canonical singularities".) These properties are independent of the choice of resolution.
Clarity¶
A clear use of Terminal singularity names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Manages Complexity¶
Terminal singularity compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—they were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata.—and the practical consequence—in particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
- Check operation and conditions. In local coordinates, this pullback operation is given by the Jacobian determinant of f.
- Demand recognition evidence. As a result, one can say more by considering these broader classes of singularities.
- Test variation.
Knowledge Transfer¶
Within the home domain. Knowledge about Terminal singularity transfers literally when a new case preserves the same carrier type, relation, and recognition test. The definitions in this paragraph are the ones used when a log resolution is not known to exist, for example when the base field has positive characteristic and X has dimension greater than 3.). This suggests that pairs can be viewed as geometric objects comparable to varieties: for some purposes, X can be replaced by the pair (Y,\Delta) . Beyond the home domain. No canonical parent is asserted for Terminal singularity.
Neighborhood in Abstraction Space¶
Terminal singularity sits in a crowded region of the domain-specific corpus (32nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- Filling radius — 0.90
- Binade — 0.88
- Julia set — 0.88
- Mehler Kernel — 0.88
- Character variety — 0.88
Computed from structural-signature embeddings · 2026-10-08