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 \S 3) has no mixed critical points on ${ \mathbb{ C}^*}^2$ (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 ${ \tilde V} \cap \hat{ \pi}^{ -1}(\boldsymbol{ 0})$ but also a real analytic manifold as a germ of ${ \tilde V} \cap \hat{ \pi}^{ -1}(\boldsymbol{ 0})$ (Theorem 15).
Categories: math.AG math.GT