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Difference between revisions of "Weierstrass point"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1985) {{MR|0770932}} {{ZBL|0559.14017}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966) {{MR|0207977}} {{ZBL|0175.36801}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1985) {{MR|0770932}} {{ZBL|0559.14017}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966) {{MR|0207977}} {{ZBL|0175.36801}} </TD></TR>
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</table>
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[[Category:Algebraic geometry]]

Revision as of 18:10, 16 October 2014

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


Comments

References

[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: http://encyclopediaofmath.org/index.php?title=Weierstrass_point&oldid=33687
This article was adapted from an original article by V.E. Voskresenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article