Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^{-}$

Abstract

We consider toric resolutions of some strongly mixed weighted homogeneous polynomials of type $J_{10}^{-}$. We show that the strongly mixed weighted homogeneous polynomial $f:=f_{2,2,1,2,1,4}(k=3)$ (see §3) has no mixed critical points on ${{\mathbb{C}}^{\ast }}^2$ (Lemma 14), and moreover, show that the strict transform $\tilde{V}$ of the mixed hypersurface singularity $V:=f^{-1}(0)$ via the toric modification $\hat{\pi }:X\to {\mathbb{C}}^2$, where we set $f:=f_{2,2,1,2,1,4}(k=3)$, is not only a real analytic manifold outside of $\tilde{V}\cap {\hat{\pi }}^{-1}(0)$ but also a real analytic manifold as a germ of $\tilde{V}\cap {\hat{\pi }}^{-1}(0)$ (Theorem 15).

15 pages