Terminal singularity¶

In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12493
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Birational Geometry, Algebraic Geometry → Mathematics

Core Idea¶

Terminal singularity is treated here as the recurring mathematics_logic_statistics 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 . Terminal singularities are important in the minimal model program because smooth minimal models do not exist in the desired generality, and hence certain "mild" singularities must be allowed.

When X is a smooth variety of dimension n over a field k, its canonical line bundle is defined as the sheaf of n-forms (or "volume forms") on X, \omega_X:=\Omega^n_{X/k} . A direct consequence of the definition of canonical singularities is that if two projective varieties with canonical singularities are birational, then they have the same plurigenera, the dimensions of the vector spaces H^0(X,O(mK_X)) for all m\geq 0 . (Over an arbitrary field, the correct statement is rather that terminal varieties of dimension at most 2 are regular schemes.) This explains why minimal models of surfaces can be taken to be smooth.

For Terminal singularity, the abstraction is narrower than the article's general subject matter: a positive case must preserve In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — 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).
Constitutive relation — They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata.
Operating condition — In local coordinates, this pullback operation is given by the Jacobian determinant of f.
Recognition evidence — As a result, one can say more by considering these broader classes of singularities.
Admissible variation — Therefore, terminal singularities in dimension 3 are isolated; over the complex numbers, they were classified by Shigefumi and Reid.
Characteristic consequence — In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Failure boundary — Over the complex numbers, they are locally analytically isomorphic to quotients of the affine plane A^2_{\mathbb C} by finite subgroups of the special linear group SL(2,{\mathbb C}) .

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Not an over-broad reading. One last related class of pairs is: (X,\Delta) is divisorial log terminal (dlt) if \Delta has coefficients at most 1 and the discrepancy is >-1 for every exceptional divisor over X whose image in X is contained in the closed subset where the pair (X,\Delta) does not have simple normal crossings.
Not an over-broad reading. 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).
Not an over-broad reading. Suppose, more strongly, that f\colon Y \to X is a log resolution, meaning that Y is nonsingular and the exceptional locus of f is a divisor with simple normal crossings in Y.
Not automatically Canonical Sheaf. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Terminal singularity applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

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 greater than 3.).
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) .
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 of singularities of X.
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).
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.
Then X is said to be. Suppose, more strongly, that f\colon Y \to X is a log resolution, meaning that Y is nonsingular and the exceptional locus of f is a divisor with simple normal crossings in Y.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: As a result, one can say more by considering these broader classes of singularities. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification One last related class of pairs is: (X,\Delta) is divisorial log terminal (dlt) if \Delta has coefficients at most 1 and the discrepancy is >-1 for every exceptional divisor over X whose image in X is contained in the closed subset where the pair (X,\Delta) does not have simple normal crossings. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Terminal singularity compresses multiple mathematics_logic_statistics 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. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics 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. Change an implementation or setting while preserving therefore, terminal singularities in dimension 3 are isolated; over the complex numbers, they were classified by Shigefumi and Reid.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For example, dx/x is a 1-form on the affine line with "log poles" at the origin.). This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group; recognition evidence → As a result, one can say more by considering these broader classes of singularities

Applied / In Practice¶

For example: the Kodaira vanishing theorem and the Cone theorem extend to projective log canonical varieties (and pairs, as discussed below) in characteristic zero. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Explanation; invariant → In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group; boundary → the case exits the class when one last related class of pairs is: (X,\Delta) is divisorial log terminal (dlt) if \Delta has coefficients at most 1 and the discrepancy is >-1 for every exceptional divisor over X whose image in X is contained in the closed subset where the pair (X,\Delta) does not have simple normal crossings

Structural Tensions¶

T1 — Stable identity versus admissible variation. One last related class of pairs is: (X,\Delta) is divisorial log terminal (dlt) if \Delta has coefficients at most 1 and the discrepancy is >-1 for every exceptional divisor over X whose image in X is contained in the closed subset where the pair (X,\Delta) does not have simple normal crossings. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. 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). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Suppose, more strongly, that f\colon Y \to X is a log resolution, meaning that Y is nonsingular and the exceptional locus of f is a divisor with simple normal crossings in Y. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. 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). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Terminal singularity literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Terminal singularity distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Terminal singularity is structural-leaning. Its structural side is the repeatable organization summarized by In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: In local coordinates, this pullback operation is given by the Jacobian determinant of f. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: 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). They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata. It further constrains recognition and variation through: In local coordinates, this pullback operation is given by the Jacobian determinant of f. As a result, one can say more by considering these broader classes of singularities.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Terminal singularity literal. Its documented scope includes the condition that 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.). Another bounded application condition is that 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) . These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Therefore, terminal singularities in dimension 3 are isolated; over the complex numbers, they were classified by Shigefumi and Reid.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Terminal singularity. The reviewed identity is: In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Terminal singularity sits in a crowded region of the domain-specific corpus (32^nd 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

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group?
Canonical Sheaf. Canonical Sheaf is a recurring identity in mathematics, logic, and statistics defined by: In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,!\Omega^n = \omega , which is the n th exterior power of the cotangent bundle \Omega on V . Tell: Which entry's carrier, operation, and failure condition are satisfied?
Pseudo-canonical variety. An algebraic variety whose canonical divisor or class is pseudo-ample under the convention used, placing it in the general-type side of birational classification. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Canonical ring. The graded ring formed from global sections of all nonnegative tensor powers of a variety's canonical bundle or canonical divisor. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Terminal singularity remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Canonical_singularity (revision 1355453133).
Preserved source candidate: http://projecteuclid.org/euclid.nmj/1118787793

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.