Weierstrass points

E898502

algebraic geometry concept point on algebraic curve

Weierstrass points are special points on an algebraic curve where the gap sequence of pole orders deviates from the generic case, reflecting deep geometric and arithmetic properties of the curve.

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Observed surface forms (1)

Surface form	Occurrences
Weierstrass point	0

Statements (48)

Predicate	Object
instanceOf	algebraic geometry concept ⓘ point on algebraic curve ⓘ
appearsIn	Hurwitz spaces NERFINISHED ⓘ Teichmüller theory NERFINISHED ⓘ
characterizedBy	deviation from generic pole order behavior ⓘ non-generic gap sequence of pole orders ⓘ
definedOn	algebraic curve ⓘ compact Riemann surface ⓘ
field	algebraic geometry ⓘ complex analysis ⓘ number theory ⓘ
generalizes	branch point of a double cover ⓘ
hasAnalogue	Weierstrass point on metric graphs ⓘ
hasInvariant	Weierstrass semigroup NERFINISHED ⓘ Weierstrass weight NERFINISHED ⓘ
hasProperty	all points are Weierstrass points on genus 0 curve ⓘ can be defined for curves over arbitrary algebraically closed fields ⓘ can be defined over non-archimedean fields ⓘ depends on genus of the curve ⓘ distribution constrained by genus and moduli ⓘ finitely many on a compact Riemann surface of genus at least 2 ⓘ no Weierstrass points on a generic elliptic curve ⓘ reflects arithmetic properties of the curve ⓘ reflects geometric properties of the curve ⓘ set is discrete on a Riemann surface ⓘ weight defined from gap sequence ⓘ
hasPropertyOnHyperellipticCurve	branch points of the hyperelliptic map are Weierstrass points GENERATED ⓘ number equals 2g+2 for genus g hyperelliptic curve GENERATED ⓘ
invariantUnder	automorphisms of the curve ⓘ
namedAfter	Karl Weierstrass NERFINISHED ⓘ
occursOn	hyperelliptic curve ⓘ
relatedTo	Baker–Norine theory on graphs NERFINISHED ⓘ Riemann–Roch theorem NERFINISHED ⓘ Weierstrass gap theorem NERFINISHED ⓘ canonical divisor ⓘ canonical embedding of a curve ⓘ divisor theory ⓘ gap sequence ⓘ holomorphic differentials ⓘ linear series on curves ⓘ pole orders of meromorphic functions ⓘ
studiedIn	Brill–Noether theory NERFINISHED ⓘ theory of Riemann surfaces ⓘ
usedIn	Arakelov theory NERFINISHED ⓘ classification of algebraic curves ⓘ coding theory on algebraic curves ⓘ moduli of curves ⓘ study of automorphism groups of curves ⓘ

Referenced by (1)

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

Brill–Noether theory → usesConcept → Weierstrass points ⓘ