Hurwitz numbers
E904001
algebraic invariant
combinatorial invariant
enumerative invariant
object of Gromov–Witten theory
object of algebraic geometry
object of enumerative geometry
object of intersection theory
object of mathematical physics
object of moduli theory
topological invariant
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hurwitz numbers canonical | 1 |
| monotone Hurwitz numbers | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11085867 — 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 numbers Context triple: [Hurwitz space, relatedTo, Hurwitz numbers]
-
A.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
-
B.
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.
-
C.
Hurwitz determinants
Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
-
D.
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.
-
E.
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.
- 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 numbers Target entity description: 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.
-
A.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
-
B.
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.
-
C.
Hurwitz determinants
Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
combinatorial invariant ⓘ enumerative invariant ⓘ object of Gromov–Witten theory ⓘ object of algebraic geometry ⓘ object of enumerative geometry ⓘ object of intersection theory ⓘ object of mathematical physics ⓘ object of moduli theory ⓘ topological invariant ⓘ |
| counts |
branched coverings of algebraic curves
ⓘ
branched coverings of the Riemann sphere ⓘ holomorphic maps between Riemann surfaces with prescribed ramification ⓘ |
| dependsOn |
degree of the covering map
ⓘ
genus of the source curve ⓘ genus of the target curve ⓘ number of branch points ⓘ ramification profile over branch points ⓘ |
| hasVariant |
ELSV-type Hurwitz numbers
ⓘ
double Hurwitz numbers ⓘ monotone Hurwitz numbers ⓘ
linked to:
Hurwitz numbers
orbifold Hurwitz numbers ⓘ quantum Hurwitz numbers ⓘ simple Hurwitz numbers ⓘ spin Hurwitz numbers ⓘ tropical Hurwitz numbers ⓘ |
| namedAfter | Adolf Hurwitz ⓘ |
| relatedTo |
2D topological gravity
ⓘ
Belyi maps ⓘ ELSV formula ⓘ Gromov–Witten invariants ⓘ Hodge integrals ⓘ KP hierarchy ⓘ
linked to:
Kadomtsev–Petviashvili equation
Riemann–Hurwitz formula ⓘ Toda hierarchy ⓘ Young diagrams ⓘ conjugacy classes in symmetric groups ⓘ dessins d’enfants ⓘ integrable hierarchies ⓘ intersection theory on moduli of curves ⓘ matrix models ⓘ moduli space of branched covers ⓘ moduli space of curves ⓘ random partitions ⓘ string theory ⓘ symmetric group representations ⓘ topological recursion ⓘ tropical geometry ⓘ |
| usedIn |
algebraic geometry
ⓘ
combinatorics of permutations ⓘ enumerative geometry ⓘ mathematical physics ⓘ |
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 numbers Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hurwitz numbers