Weierstrass point

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


[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



[a1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978)
[a2] E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985)
[a3] R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966)
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