Hurwitz surfaces
E907201
Hurwitz surfaces are compact Riemann surfaces of maximal possible automorphism group size for their genus, achieving the Hurwitz bound of 84(g−1) symmetries.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hurwitz surfaces canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11098652 — 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: Hurwitz surfaces Context triple: [(2,3,7) triangle group, isAssociatedWith, Hurwitz surfaces]
-
A.
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.
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B.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
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C.
Hurwitz numbers
Hurwitz numbers are algebraic invariants that count branched coverings of the Riemann sphere (or other curves) with specified ramification data, playing a key role in enumerative geometry and mathematical physics.
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D.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
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E.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
- 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: Hurwitz surfaces Target entity description: Hurwitz surfaces are compact Riemann surfaces of maximal possible automorphism group size for their genus, achieving the Hurwitz bound of 84(g−1) symmetries.
-
A.
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.
-
B.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
-
C.
Hurwitz numbers
Hurwitz numbers are algebraic invariants that count branched coverings of the Riemann sphere (or other curves) with specified ramification data, playing a key role in enumerative geometry and mathematical physics.
-
D.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
-
E.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hurwitz surface
ⓘ
Hurwitz surface ⓘ Hurwitz surface ⓘ algebraic curve ⓘ compact Riemann surface ⓘ complex algebraic curve ⓘ geometric object ⓘ |
| characterizedBy | |Aut(X)| = 84(g(X)−1) ⓘ |
| correspondsTo | points with maximal automorphism group in moduli space of curves ⓘ |
| definedOver | complex numbers ⓘ |
| hasAutomorphismGroup |
Hurwitz group
NERFINISHED
ⓘ
PSL(2,7) NERFINISHED ⓘ finite group ⓘ group of conformal automorphisms ⓘ |
| hasAutomorphismGroupOrder |
168
ⓘ
504 ⓘ 84(g−1) ⓘ |
| hasAutomorphismGroupOrderUpperBound | 84(g−1) ⓘ |
| hasBelyiMap | ramified over three points with ramification type (2,3,7) ⓘ |
| hasCurvature | negative ⓘ |
| hasEulerCharacteristic | 2−2g ⓘ |
| hasExample |
Fricke–Macbeath curve
NERFINISHED
ⓘ
Klein quartic NERFINISHED ⓘ Macbeath surface NERFINISHED ⓘ |
| hasFundamentalGroup | torsion-free subgroup of (2,3,7) triangle group ⓘ |
| hasGenus |
3
ⓘ
7 ⓘ g ≥ 2 ⓘ |
| hasMetric | hyperbolic metric ⓘ |
| hasProperty | maximal automorphism group for its genus ⓘ |
| hasSymmetryType | maximally symmetric for its genus ⓘ |
| is | quotient of hyperbolic plane by a torsion-free subgroup of (2,3,7) triangle group ⓘ |
| isQuotientOf | hyperbolic plane ⓘ |
| liesIn | moduli space of curves of genus g ⓘ |
| maximizes | number of conformal automorphisms for given genus ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| relatedTo |
Fuchsian group
ⓘ
Galois Belyi map ⓘ Hurwitz group NERFINISHED ⓘ Hurwitz’s automorphism theorem NERFINISHED ⓘ triangle group (2,3,7) NERFINISHED ⓘ |
| satisfies | Hurwitz bound ⓘ |
| studiedIn |
Riemann surface theory
ⓘ
algebraic geometry ⓘ complex analysis ⓘ geometric group theory ⓘ |
| uniformizedBy | (2,3,7) triangle group NERFINISHED ⓘ |
| usedIn | construction of large 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: Hurwitz surfaces Description of subject: Hurwitz surfaces are compact Riemann surfaces of maximal possible automorphism group size for their genus, achieving the Hurwitz bound of 84(g−1) symmetries.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.