Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^-$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910350049280000 |
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| author | Saito, Sachiko |
| author_facet | Saito, Sachiko |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^-$ Saito, Sachiko Algebraic Geometry Geometric Topology 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). |
| title | Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^-$ |
| topic | Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2402.15660 |