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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2605.07493 |
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| _version_ | 1866909025902264320 |
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| author | Tumenbayar, Khulan |
| author_facet | Tumenbayar, Khulan |
| contents | Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C} \cap \mathcal{Q}$ is even and $\mathcal{C} \cap \mathcal{Q}$ contain at least one smooth point $z_o$ of $\mathcal{Q}$. In this paper, for every $\mathcal{Q}$ we find all possible conics $\mathcal{C}$ as above via studying geometry of $\mathcal{C}$ and $\mathcal{Q}$ through that of integral sections of a rational elliptic surface which canonically arises from $\mathcal{Q}$ and $z_o \in \mathcal{C} \cap \mathcal{Q}$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mathcal{Q}$, $\mathcal{C}$ and line passing through two of the singular points of $\mathcal{Q}$ . |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs Tumenbayar, Khulan Algebraic Geometry 14J27, 14H50 Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C} \cap \mathcal{Q}$ is even and $\mathcal{C} \cap \mathcal{Q}$ contain at least one smooth point $z_o$ of $\mathcal{Q}$. In this paper, for every $\mathcal{Q}$ we find all possible conics $\mathcal{C}$ as above via studying geometry of $\mathcal{C}$ and $\mathcal{Q}$ through that of integral sections of a rational elliptic surface which canonically arises from $\mathcal{Q}$ and $z_o \in \mathcal{C} \cap \mathcal{Q}$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mathcal{Q}$, $\mathcal{C}$ and line passing through two of the singular points of $\mathcal{Q}$ . |
| title | Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs |
| topic | Algebraic Geometry 14J27, 14H50 |
| url | https://arxiv.org/abs/2605.07493 |