Weierstrass point

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2010 Mathematics Subject Classification: Primary: 14H55 [MSN][ZBL]

A point on an algebraic curve (or on a Riemann surface) $X$ of genus $g$ at which the following condition is satisfied: There exists a non-constant rational function on $X$ which has at this point a pole of order not exceeding $g$ and which has no singularities at other points of $X$. Only a finite number of Weierstrass points can exist on $X$, and if $g$ is 0 or 1, there are no such points at all, while if $g\geq2$, Weierstrass points must exist. These results were obtained for Riemann surfaces by K. Weierstrass. For algebraic curves of genus $g\geq2$ there always exist at least $2g+2$ Weierstrass points, and only hyper-elliptic curves of genus $g$ have exactly $2g+2$ Weierstrass points. The upper bound on the number of Weierstrass points is $(g-1)g(g+1)$. The presence of a Weierstrass point on an algebraic curve $X$ of genus $g\geq2$ ensures the existence of a morphism of degree not exceeding $g$ from the curve $X$ onto the projective line $P^1$.


[1] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)
[2] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602



[a1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2] E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) MR0770932 Zbl 0559.14017
[a3] R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966) MR0207977 Zbl 0175.36801
How to Cite This Entry:
Weierstrass point. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Voskresenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article