Halphen pencil

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)