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.

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 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 (2)

Subject (surface form when different)	Predicate
Riemann surface →	hasTheorem
Bernhard Riemann →	knownFor