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 ${\mathbb{C}^*}^2$ (Lemma 14), and moreover, show that the strict transform $\tilde V$ of the mixed hypersurface singularity $V := f^{-1}(0)$ via the toric modification $\hatπ : X \to \mathbb{C}^2$, where we set $f := f_{2,2,1,2,1,4}\ (k=3)$, is not only a real analytic manifold outside of ${\tilde V} \cap \hatπ^{-1}(\boldsymbol{0})$ but also a real analytic manifold as a germ of ${\tilde V} \cap \hatπ^{-1}(\boldsymbol{0})$ (Theorem 15).

Figures (4)

• Figure 1: The isotopy types of nonsingular real algebraic $M$-curves of degree $6$ on $\mathbb{R} P^2$
• Figure 2: The radial Newton polyhedron of $f_{a,b,c,d,e,\mathrm{f}}$
• Figure 3: The dual Newton diagram $\Gamma^*(f)$
• Figure 4: The regular simplicial subdivision $\Sigma^*$ which is admissible for $\Gamma^*(f)$.

Theorems & Definitions (17)

• Definition 1
• Definition 2: Oka2018, p.79
• Definition 3: Oka2018, p.182; see also Oka2015
• Example 4: Oka2018, Example 9.17
• Lemma 5: cf.Oka2008, Oka2010
• Proposition 6: Euler equalities, Oka2018
• Definition 7: mixed weighted homogeneous polynomial, Oka2018, pp.182--184
• Definition 8: cf.Oka2018,Definition 9.18; Oka2015,p.174
• Definition 9: Oka2010, p.6, Definition 3; Oka2018, p.80 and pp.181--182
• Definition 10: Oka2018, p.80 and p.182