# Weierstrass point

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 $P^1$.

#### 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 MR0092855 Zbl 0078.06602