Riemann–Hurwitz formula

E47610

mathematical theorem result in algebraic geometry result in complex analysis

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.

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All labels observed (1)

Label	Occurrences
Riemann–Hurwitz formula canonical	6

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in algebraic geometry ⓘ result in complex analysis ⓘ
appearsIn	textbooks on Riemann surfaces ⓘ textbooks on algebraic curves ⓘ textbooks on algebraic geometry ⓘ
appliesTo	branched covering of Riemann surfaces ⓘ finite holomorphic maps between compact Riemann surfaces ⓘ finite morphisms of smooth projective algebraic curves ⓘ
assumes	compact connected Riemann surfaces ⓘ finite degree d of the covering map ⓘ holomorphic surjective map between Riemann surfaces ⓘ
describes	effect of branched coverings on genus ⓘ relationship between genera of Riemann surfaces ⓘ
field	algebraic curves ⓘ algebraic geometry ⓘ complex analysis ⓘ theory of Riemann surfaces ⓘ
generalizationOf	Euler characteristic multiplicativity for unramified coverings ⓘ
gives	formula for Euler characteristic under branched covering ⓘ formula for genus of a covering curve ⓘ
hasForm	2g(X) - 2 = d(2g(Y) - 2) + sum_{x in X}(e_x - 1) ⓘ
historicalPeriod	19th century mathematics ⓘ
involvesConcept	Euler’s polyhedron formula ⓘ surface form: Euler characteristic branch point ⓘ degree of a covering map ⓘ genus of a Riemann surface ⓘ holomorphic map ⓘ ramification index ⓘ topological covering space ⓘ
namedAfter	Adolf Hurwitz ⓘ Bernhard Riemann ⓘ
relatedTo	Grothendieck–Ogg–Shafarevich formula ⓘ Hurwitz bound on automorphism groups of curves ⓘ Hurwitz space ⓘ Lefschetz fixed-point theorem ⓘ covering space theory ⓘ
relates	genus of domain Riemann surface ⓘ genus of target Riemann surface ⓘ ramification data of a covering map ⓘ
usedFor	classifying algebraic curves by coverings ⓘ computing genus of algebraic curves ⓘ computing genus of function fields extensions ⓘ studying ramified coverings ⓘ
usedIn	Galois covers of curves ⓘ arithmetic geometry ⓘ moduli theory of curves ⓘ number theory ⓘ theory of algebraic function fields ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Bernhard Riemann → knownFor → Riemann–Hurwitz formula ⓘ

Riemann surfaces → hasTheorem → Riemann–Hurwitz formula ⓘ

subject surface form: Riemann surface

Hurwitz space → relatedTo → Riemann–Hurwitz formula ⓘ

Hurwitz bound on automorphism groups of curves → proofUses → Riemann–Hurwitz formula ⓘ

Grothendieck–Ogg–Shafarevich formula → isRelatedTo → Riemann–Hurwitz formula ⓘ

Adolf Hurwitz → knownFor → Riemann–Hurwitz formula ⓘ