Triple
T10772687
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Grothendieck–Riemann–Roch theorem |
E254119
|
entity |
| Predicate | hasConsequence |
P812
|
FINISHED |
| Object | Hirzebruch–Riemann–Roch for smooth projective varieties |
E259772
|
NE FINISHED |
How this triple was built (2 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: Hirzebruch–Riemann–Roch for smooth projective varieties | Statement: [Grothendieck–Riemann–Roch theorem, hasConsequence, Hirzebruch–Riemann–Roch for smooth projective varieties]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hirzebruch–Riemann–Roch for smooth projective varieties Context triple: [Grothendieck–Riemann–Roch theorem, hasConsequence, Hirzebruch–Riemann–Roch for smooth projective varieties]
-
A.
Hirzebruch–Riemann–Roch theorem
chosen
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
B.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
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.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
E.
Lazarsfeld’s Positivity in Algebraic Geometry
Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
- F. None of above.
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Provenance (3 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_69d6aa5f54f4819082d0bbcb6f8797e6 |
completed | April 8, 2026, 7:19 p.m. |
| NER | Named-entity recognition | batch_69d7329b27748190bd0e2569c7972fd1 |
completed | April 9, 2026, 5:01 a.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69de238559b48190abc759e744ab0f8e |
completed | April 14, 2026, 11:22 a.m. |
Created at: April 8, 2026, 9:16 p.m.