A Eudoxian study of discriminant curves associated to normal surface singularities
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918385028169728 |
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| author | Barroso, Evelia Rosa García Popescu-Pampu, Patrick |
| author_facet | Barroso, Evelia Rosa García Popescu-Pampu, Patrick |
| contents | Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial Newton polynomial of a defining series of the discriminant germ of $(f,g)$ depends up to toric automorphisms only on the germs of curves defined by $f$ and $g$. This result generalizes a theorem of Gryszka, Gwoździewicz and Parusiński, which is the special case in which $(S,s)$ is smooth. Our proof uses a common generalization of formulas of Lê, Casas-Alvero and Némethi for the intersection number of the discriminant with a germ of plane curve. It uses also a theorem of Delgado and Maugendre characterizing the special members of pencils of curves on normal surface singularities. We apply it to the pencils generated by all pairs $(f^b, g^a)$, for varying positive integral exponents $a, b$, following a strategy initiated by Gwoździewicz and by Delgado and Maugendre. This is similar to the Eudoxian method of comparison of magnitudes by comparing the sizes of their positive integral multiples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Eudoxian study of discriminant curves associated to normal surface singularities Barroso, Evelia Rosa García Popescu-Pampu, Patrick Algebraic Geometry 14B05, 14H20 Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial Newton polynomial of a defining series of the discriminant germ of $(f,g)$ depends up to toric automorphisms only on the germs of curves defined by $f$ and $g$. This result generalizes a theorem of Gryszka, Gwoździewicz and Parusiński, which is the special case in which $(S,s)$ is smooth. Our proof uses a common generalization of formulas of Lê, Casas-Alvero and Némethi for the intersection number of the discriminant with a germ of plane curve. It uses also a theorem of Delgado and Maugendre characterizing the special members of pencils of curves on normal surface singularities. We apply it to the pencils generated by all pairs $(f^b, g^a)$, for varying positive integral exponents $a, b$, following a strategy initiated by Gwoździewicz and by Delgado and Maugendre. This is similar to the Eudoxian method of comparison of magnitudes by comparing the sizes of their positive integral multiples. |
| title | A Eudoxian study of discriminant curves associated to normal surface singularities |
| topic | Algebraic Geometry 14B05, 14H20 |
| url | https://arxiv.org/abs/2410.21250 |