The stratum with constant milnor number of a mini-transversal family of a quasihomogeneous function of corank two

Abstract

$1. INTRODUCTlON LET~:(C”, O)-+(C, 0) be a germ of quasihomogeneous function of Milnor number p with an isolated critical point. Let F:(C” x C’-‘, O)+(C, 0) be transversal to the orbit offin the space of germs of holomorphic functions preserving the origin with an isolated critical point, where we have the orbits of the action of the group of germs of biholomorphic mappings preserving the origin. We call the family F, a mini-transversal family off and call the stratum with constant Milnor number of F, the germ at the origin of the set of those values of parameters t for which F, has an isolated critical point at the origin of the same Milnor number asf. In [I], Amol’d showed that the stratum with constant Milnor number of a minitransversal family contains a germ of non-singular algebraic subset in (?‘-I of dimension m(f), where m(f) is the number of generators of a monomial basis of the finitely dimensional C-vector space C(x,, . . . , x,}/(8f/&,, . . . , c?~/c?x,) above and on the Newton boundary of 5 We call the number m(f) the inner modarity off (see [ 11). In [ 11, he conjectured that these germs coincide and showed this conjecture for 0and l-modal quasihomogeneous functions (see [l, 21). In [13], we showed it for 2-modal quasihomogeneous functions (see [l, 21). In [4], Gabrielov and Kushnirenko showed it for homogeneous functions, using the result of L& Dung Trang and Saito[7J The author does not know of any further work on this problem. In this paper, we shall determine the stratum with constant Milnor number of a mini-transversal family of a quasihomogeneous function of corank two with an isolated critical point; to prove the conjecture of Arnol’d for them. The details of our results are the following