# Weierstrass point

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 .

#### References

[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 |

#### Comments

#### References

[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: http://encyclopediaofmath.org/index.php?title=Weierstrass_point&oldid=12917