Hurwitz space
E262118
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hurwitz space canonical | 2 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
geometric object
ⓘ
moduli space ⓘ parameter space ⓘ |
| canBe |
coarse moduli space
ⓘ
fine moduli space in suitable settings ⓘ |
| dependsOn |
degree of the covering
ⓘ
genus of the source curve ⓘ genus of the target curve ⓘ monodromy group ⓘ ramification profile ⓘ |
| describes | isomorphism classes of branched covers with fixed branching data ⓘ |
| field |
Teichmüller theory
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| generalizes | classical Hurwitz schemes ⓘ |
| hasConcept |
branch morphism to configuration space of branch points
ⓘ
connected components classified by monodromy data ⓘ |
| hasInvariant | dimension determined by Riemann–Hurwitz formula ⓘ |
| hasProperty |
may have boundary corresponding to degenerate covers
ⓘ
often constructed as a complex analytic space ⓘ often constructed as an algebraic variety ⓘ often quasi-projective ⓘ specified branching data ⓘ |
| namedAfter | Adolf Hurwitz ⓘ |
| oftenEquippedWith |
algebraic structure
ⓘ
natural complex structure ⓘ universal family of covers ⓘ |
| parametrizes |
branched covers of Riemann surfaces
ⓘ
branched covers of algebraic curves ⓘ |
| relatedTo |
Belyi maps
ⓘ
Galois covers ⓘ Hurwitz numbers ⓘ Riemann–Hurwitz formula ⓘ braid group actions ⓘ configuration space of points on a curve ⓘ mapping class group ⓘ moduli space of curves ⓘ |
| studiedInContextOf |
deformation theory of covers
ⓘ
stable reduction of covers ⓘ |
| usedIn |
Galois theory of function fields
ⓘ
enumerative geometry ⓘ inverse Galois problem ⓘ study of branched coverings ⓘ topology of surface bundles ⓘ |
| usedToStudy |
arithmetic of function fields
ⓘ
distribution of Galois groups of covers ⓘ specialization of covers ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.