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) |
Weierstrass point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_point&oldid=24009