## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 47, Number 1 (2018), 223-244.

### Elliptic surfaces and contact conics for a 3-nodal quartic

Khulan TUMENBAYAR and Hiro-o TOKUNAGA

#### Abstract

Let ${\mathcal Q}$ be an irreducible $3$-nodal quartic and let ${\mathcal C}$ be a smooth conic such that ${\mathcal C} \cap {\mathcal Q}$ does not contain any node of ${\mathcal Q}$ and the intersection multiplicity at $z \in {\mathcal C} \cap {\mathcal Q}$ is even for each $z$. In this paper, we study geometry of ${\mathcal C} + {\mathcal Q}$ through that of integral sections of a rational elliptic surface which canonically arises from ${\mathcal Q}$ and $z \in {\mathcal C} \cap {\mathcal Q}$. As an application, we construct Zariski pairs $({\mathcal C}_1 + {\mathcal Q}, {\mathcal C}_2 + {\mathcal Q})$, where ${\mathcal C}_i$ $(i = 1, 2)$ are smooth conics tangent to ${\mathcal Q}$ at four distinct points.

#### Article information

**Source**

Hokkaido Math. J., Volume 47, Number 1 (2018), 223-244.

**Dates**

First available in Project Euclid: 13 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1520928068

**Digital Object Identifier**

doi:10.14492/hokmj/1520928068

**Mathematical Reviews number (MathSciNet)**

MR3773733

**Zentralblatt MATH identifier**

06853599

**Subjects**

Primary: 14J27: Elliptic surfaces 14H30: Coverings, fundamental group [See also 14E20, 14F35] 14H50: Plane and space curves

**Keywords**

Elliptic surface section contact conic Zariski pair

#### Citation

TUMENBAYAR, Khulan; TOKUNAGA, Hiro-o. Elliptic surfaces and contact conics for a 3-nodal quartic. Hokkaido Math. J. 47 (2018), no. 1, 223--244. doi:10.14492/hokmj/1520928068. https://projecteuclid.org/euclid.hokmj/1520928068