Triple
T17386212
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Oscar Zariski |
E422693
|
entity |
| Predicate | knownFor |
P22
|
FINISHED |
| Object |
Zariski’s Main Theorem
Zariski’s Main Theorem is a fundamental result in algebraic geometry that characterizes finite-type morphisms between varieties by relating birationality, normality, and the structure of fibers.
|
E1266004
|
NE FINISHED |
How this triple was built (4 steps)
Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.
NER
Named-entity recognition
gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Zariski’s Main Theorem | Statement: [Oscar Zariski, knownFor, Zariski’s Main Theorem]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Zariski’s Main Theorem Context triple: [Oscar Zariski, knownFor, Zariski’s Main Theorem]
-
A.
Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
-
B.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex geometry.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Grothendieck–Lefschetz theorem
The Grothendieck–Lefschetz theorem is a fundamental result in algebraic geometry that extends Lefschetz-type hyperplane theorems to a broad scheme-theoretic and cohomological setting, relating the geometry and Picard groups of a variety to those of its hyperplane sections.
-
E.
Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg
Description generation
gpt-5.1
Instruction
Generate a one-sentence description of the target entity. You are given a context triple in the form (subject, predicate, object), where the object is the target entity. # Instructions Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. Avoid repeating the information from the triple, unless really essential. # Response Format Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Zariski’s Main Theorem Triple: [Oscar Zariski, knownFor, Zariski’s Main Theorem]
Generated description
Zariski’s Main Theorem is a fundamental result in algebraic geometry that characterizes finite-type morphisms between varieties by relating birationality, normality, and the structure of fibers.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Zariski’s Main Theorem Target entity description: Zariski’s Main Theorem is a fundamental result in algebraic geometry that characterizes finite-type morphisms between varieties by relating birationality, normality, and the structure of fibers.
-
A.
Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
-
B.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex geometry.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Grothendieck–Lefschetz theorem
The Grothendieck–Lefschetz theorem is a fundamental result in algebraic geometry that extends Lefschetz-type hyperplane theorems to a broad scheme-theoretic and cohomological setting, relating the geometry and Picard groups of a variety to those of its hyperplane sections.
-
E.
Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
- F. None of above. chosen
Provenance (5 batches)
The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.
| Step | Stage | Batch ID | Status | When |
|---|---|---|---|---|
| creating | Elicitation | batch_69d889d710288190bf0f4762801fefae |
completed | April 10, 2026, 5:25 a.m. |
| NER | Named-entity recognition | batch_69e43a89c5008190a277a68e5cfe67b7 |
completed | April 19, 2026, 2:14 a.m. |
| NED1 | Entity disambiguation (via context triple) | batch_6a019ff95e7c81909564abc3fe319611 |
completed | May 11, 2026, 9:23 a.m. |
| NEDg | Description generation | batch_6a01a0c2f1108190b3cdabdcd3da7253 |
completed | May 11, 2026, 9:26 a.m. |
| NED2 | Entity disambiguation (via description) | batch_6a01a14aeab881908abf4ec28164ba84 |
completed | May 11, 2026, 9:28 a.m. |
Created at: April 10, 2026, 5:45 a.m.