Geometry of cuspidal sextics and their dual curves

IntroductionLet C be a given irreducible plane curve of degree n defined by f ( x, y) = 0 where f ( x, y) is an irreducible polynomial.C is called a torus curve of type (p,q) if p,qln and f(x,y) is written as f(x,y) = f njp(x, y)P + f n;q(x, y)q for some polynomials f n/p, f n/q of degree n/p and n/q respectively.This terminology is due to Kulikov, [K2].Torus curves have been studied by many authors (In the process of studying Zariski pairs in the moduli of plane curves of degree 6 with 3 cusps of type y 4 -x 3 = 0, we have observed that there exist two irreducible components N3,i/ PSL(3, C) and N3,2/ PSL(3, C) which corresponds to torus curves and non-torus curves respectively (Lemma 25).Their dual curves are sextics with six cusps and three nodes.Starting from this observation, we study the moduli space of sextic with 6 cusps and 3 nodes which we denote by M and the moduli of their dual curves.It turns out that M has a beautiful symmetry.The "regular part" (=Plucker curves) of M is stable by the dual curve operation and the moduli of 3 (3,4)-cuspidal sextics N3 is on the "boundary" of M in a nice way (Theorem 18).By the dual operation, this moduli is isomorphic to a "singular" stratum M 3 of M, which consists of 6 cuspidal 3 nodal sextics with 3 fl.exes of order 2. The moduli space M is a disjoint union of torus curves and non-torus curves.The generic Alexander polynomial ~c(t) of P 2 -C is determined by the type of C. Namely if C is a torus curve, ~c(t) = t 2 -t + I and 1r1 (P 2 -C) = Z2 * Z3, while for non-torus curve C, ~c ( t) = 1.Moreover we show that the dual curve C* is a torus curve if and only if C is a torus curve.This is striking, as it implies also that the topology of the complement is preserved by the dual operation for a torus curve in M.