Weierstrass points
E898502
Weierstrass points are special points on an algebraic curve where the gap sequence of pole orders deviates from the generic case, reflecting deep geometric and arithmetic properties of the curve.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Weierstrass gap theorem | 1 |
| Weierstrass points canonical | 1 |
| Weierstrass semigroup | 1 |
| Weierstrass weight | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992017 — 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: Weierstrass points Context triple: [Brill–Noether theory, usesConcept, Weierstrass points]
-
A.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
B.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
C.
Weierstrass form
Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
-
D.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
-
E.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
- 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: Weierstrass points Target entity description: Weierstrass points are special points on an algebraic curve where the gap sequence of pole orders deviates from the generic case, reflecting deep geometric and arithmetic properties of the curve.
-
A.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
B.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
C.
Weierstrass form
Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
-
D.
Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
-
E.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic geometry concept
ⓘ
point on algebraic curve ⓘ |
| appearsIn |
Hurwitz spaces
ⓘ
linked to:
Hurwitz space
Teichmüller theory ⓘ |
| characterizedBy |
deviation from generic pole order behavior
ⓘ
non-generic gap sequence of pole orders ⓘ |
| definedOn |
algebraic curve
ⓘ
compact Riemann surface ⓘ |
| field |
algebraic geometry
ⓘ
complex analysis ⓘ number theory ⓘ |
| generalizes | branch point of a double cover ⓘ |
| hasAnalogue | Weierstrass point on metric graphs ⓘ |
| hasInvariant |
Weierstrass semigroup
ⓘ
linked to:
Weierstrass points
Weierstrass weight ⓘ
linked to:
Weierstrass points
|
| hasProperty |
all points are Weierstrass points on genus 0 curve
ⓘ
can be defined for curves over arbitrary algebraically closed fields ⓘ can be defined over non-archimedean fields ⓘ depends on genus of the curve ⓘ distribution constrained by genus and moduli ⓘ finitely many on a compact Riemann surface of genus at least 2 ⓘ no Weierstrass points on a generic elliptic curve ⓘ reflects arithmetic properties of the curve ⓘ reflects geometric properties of the curve ⓘ set is discrete on a Riemann surface ⓘ weight defined from gap sequence ⓘ |
| hasPropertyOnHyperellipticCurve |
branch points of the hyperelliptic map are Weierstrass points
ⓘ
number equals 2g+2 for genus g hyperelliptic curve ⓘ |
| invariantUnder | automorphisms of the curve ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| occursOn | hyperelliptic curve ⓘ |
| relatedTo |
Baker–Norine theory on graphs
ⓘ
Riemann–Roch theorem ⓘ Weierstrass gap theorem ⓘ
linked to:
Weierstrass points
canonical divisor ⓘ canonical embedding of a curve ⓘ divisor theory ⓘ gap sequence ⓘ holomorphic differentials ⓘ linear series on curves ⓘ pole orders of meromorphic functions ⓘ |
| studiedIn |
Brill–Noether theory
ⓘ
theory of Riemann surfaces ⓘ |
| usedIn |
Arakelov theory
ⓘ
classification of algebraic curves ⓘ coding theory on algebraic curves ⓘ moduli of curves ⓘ study of automorphism groups of 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: Weierstrass points Description of subject: Weierstrass points are special points on an algebraic curve where the gap sequence of pole orders deviates from the generic case, reflecting deep geometric and arithmetic properties of the curve.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Weierstrass points
linked to: Weierstrass points
subject linked to:
Weierstrass points
linked to: Weierstrass points
subject linked to:
Weierstrass points
linked to: Weierstrass points