Plücker formulas
E291201
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Plücker formulas canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2689819 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Plücker formulas Context triple: [Julius Plücker, notableWork, Plücker formulas]
-
A.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
-
B.
Clebsch diagonal surfaces
Clebsch diagonal surfaces are classical 19th-century algebraic surfaces in projective three-space, famous as the first explicit smooth cubic surface with all 27 lines defined over the real numbers.
-
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.
Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Plücker formulas Target entity description: Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
-
A.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
-
B.
Clebsch diagonal surfaces
Clebsch diagonal surfaces are classical 19th-century algebraic surfaces in projective three-space, famous as the first explicit smooth cubic surface with all 27 lines defined over the real numbers.
-
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.
Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
classical result in algebraic geometry
ⓘ
set of formulas in algebraic geometry ⓘ |
| appearsIn | classical textbooks on algebraic geometry ⓘ |
| appliesTo |
irreducible plane projective curves
ⓘ
plane algebraic curves ⓘ |
| associatedWith |
enumerative geometry
ⓘ
projective duality ⓘ |
| assumes | curve is reduced ⓘ |
| context | projective plane ⓘ |
| expresses | constraints on possible singularity configurations ⓘ |
| field | algebraic geometry ⓘ |
| generalizationOf | earlier results on plane curve singularities ⓘ |
| givesRelationBetween |
degree and class of a plane curve
ⓘ
degree of a curve and degree of its dual ⓘ singularities of a curve and invariants of its dual ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| influenced |
development of classical projective geometry
ⓘ
later work on dual varieties and discriminants ⓘ |
| involves |
class of a plane curve
ⓘ
class of the dual curve ⓘ degree of the dual curve ⓘ dual curve of a plane curve ⓘ number of cusps of a plane curve ⓘ number of nodes of a plane curve ⓘ |
| mathematicalDomain |
complex algebraic curves
ⓘ
projective algebraic curves ⓘ |
| namedAfter | Julius Plücker ⓘ |
| relatedConcept |
Severi varieties
ⓘ
adjunction formula ⓘ dual variety ⓘ genus of a plane curve ⓘ |
| relates |
degree of a plane curve
ⓘ
invariants of the dual curve ⓘ singularities of a plane curve ⓘ |
| requires |
classification of plane curve singularities into nodes and cusps
ⓘ
intersection theory in the projective plane ⓘ notions of multiplicity of intersection ⓘ |
| usedFor |
classifying plane algebraic curves
ⓘ
computing invariants of dual curves ⓘ studying singularities of plane curves ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Plücker formulas Description of subject: Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.