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A pencil of plane algebraic curves of degree $3n$ with nine $n$-tuple basis points. Such pencils were first studied by G. Halphen [[#References|[1]]] for $n=2$. The basis points $P_1,\dots,P_9$ of a Halphen pencil, which may also include infinitely near points, always lie on a cubic curve $F=F(x_0,x_1,x_2)=0$. An arbitrary curve from the Halphen pencil has the equation $\lambda G+\mu F^n=0$, where $G=G(x_0,x_1,x_2)=0$ is an elliptic curve of degree $3n$ with singular points $P_1,\dots,P_9$ of multiplicity $n$. If $F=0$ is a non-singular curve, then, with respect to the group law on this curve, $n(P_1\oplus\ldots\oplus P_9)=0$. This fact can be generalized to the case when $F=0$ is a curve with singular points [[#References|[3]]]. Each pencil of elliptic curves on a plane may be transformed, by a birational transformation of the plane, into a Halphen pencil [[#References|[2]]], [[#References|[3]]].
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A pencil of plane algebraic curves of degree $3n$ with nine $n$-tuple basis points. Such pencils were first studied by G. Halphen [[#References|[1]]] for $n=2$. The basis points $P_1,\dotsc,P_9$ of a Halphen pencil, which may also include infinitely near points, always lie on a cubic curve $F=F(x_0,x_1,x_2)=0$. An arbitrary curve from the Halphen pencil has the equation $\lambda G+\mu F^n=0$, where $G=G(x_0,x_1,x_2)=0$ is an elliptic curve of degree $3n$ with singular points $P_1,\dotsc,P_9$ of multiplicity $n$. If $F=0$ is a non-singular curve, then, with respect to the group law on this curve, $n(P_1\oplus\dotsb\oplus P_9)=0$. This fact can be generalized to the case when $F=0$ is a curve with singular points [[#References|[3]]]. Each pencil of elliptic curves on a plane may be transformed, by a birational transformation of the plane, into a Halphen pencil [[#References|[2]]], [[#References|[3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Halphen,  "Sur les courbes planes du sixième degré à neuf points double"  ''Bull. Soc. Math. France'' , '''10'''  (1882)  pp. 162–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Bertini,  ''Ann. Mat. Pura Appl.'' , '''8'''  (1877)  pp. 224–286</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.V. Dolgachev,  "On rational surfaces with a pencil of elliptic curves"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 1073–1100  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Halphen,  "Sur les courbes planes du sixième degré à neuf points double"  ''Bull. Soc. Math. France'' , '''10'''  (1882)  pp. 162–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Bertini,  ''Ann. Mat. Pura Appl.'' , '''8'''  (1877)  pp. 224–286</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.V. Dolgachev,  "On rational surfaces with a pencil of elliptic curves"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 1073–1100  (In Russian)</TD></TR></table>

Latest revision as of 12:52, 14 February 2020

A pencil of plane algebraic curves of degree $3n$ with nine $n$-tuple basis points. Such pencils were first studied by G. Halphen [1] for $n=2$. The basis points $P_1,\dotsc,P_9$ of a Halphen pencil, which may also include infinitely near points, always lie on a cubic curve $F=F(x_0,x_1,x_2)=0$. An arbitrary curve from the Halphen pencil has the equation $\lambda G+\mu F^n=0$, where $G=G(x_0,x_1,x_2)=0$ is an elliptic curve of degree $3n$ with singular points $P_1,\dotsc,P_9$ of multiplicity $n$. If $F=0$ is a non-singular curve, then, with respect to the group law on this curve, $n(P_1\oplus\dotsb\oplus P_9)=0$. This fact can be generalized to the case when $F=0$ is a curve with singular points [3]. Each pencil of elliptic curves on a plane may be transformed, by a birational transformation of the plane, into a Halphen pencil [2], [3].

References

[1] G.H. Halphen, "Sur les courbes planes du sixième degré à neuf points double" Bull. Soc. Math. France , 10 (1882) pp. 162–172
[2] E. Bertini, Ann. Mat. Pura Appl. , 8 (1877) pp. 224–286
[3] I.V. Dolgachev, "On rational surfaces with a pencil of elliptic curves" Izv. Akad. Nauk SSSR Ser. Mat. , 30 (1966) pp. 1073–1100 (In Russian)
How to Cite This Entry:
Halphen pencil. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Halphen_pencil&oldid=44603
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article