Geometry of sextics and their dual curves
by
Mutsuo Oka
Tokyo Metropolitan University

In this talk, we will explain some interesting geometry of cuspidal sextics. We consider the moduli space M of sextics with six cusps and three nodes. It is self-dual by the dual curve operation. The submoduli of curves of torus-type is denoted by M_torus. We will show that:

Theorem 1.

1. M is self-dual by the dual curve operation. Furthermore, the dual operation preserves curves of torus-type and non-torus-type. For C in M, C is in M_torus iff \Delta_C(t)=t²-t+1.

2. Let [^(M)]_torus be the connected component of M(24\beta_{2, 3}+24\beta_{2, 2};12) which contains a curve of torus-type. Then, [^(M)] consists of (maybe not all) quasi-torus curves and [^(M)]_torus is also invariant under the * operation. Their Alexander polynomials are given by t²-t+1.

3. There exists a canonical morphism \psi: M_torus --> [^(M)]_torus and an involution \iota:M_torus --> M_torus (\iota =/= * ) such that the diagram commutes:

Mtorus \iota -->   Mtorus \downarrow\rlap\psi \downarrow\rlap\psi ^ M   torus  * -->   ^ M   torus

Theorem 2. (Self-dual three (3, 4)-cuspidal sextics).

There exists a unique self-dual sextic with three (3, 4)-cusps. It is of torus-type and is given by f:=f₂³+54f₃², where f₂:=y²-2x+x²,     f₃:=(y²-x²)(x-1).

http://www.comp.metro-u.ac.jp/~oka

Date received: June 4, 1999