Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs

TL;DR

This paper classifies certain conics tangent to a special quartic curve with singularities using rational elliptic surfaces, and constructs Zariski pairs with specific geometric configurations.

Contribution

It introduces a method to find all such conics via elliptic surface geometry and constructs explicit Zariski pairs involving these configurations.

Findings

Classified all conics with even intersection multiplicity with the quartic.

Connected the geometry of these conics to integral sections of a rational elliptic surface.

Constructed Zariski pairs of degrees 7 and 8 with specified singularities.

Abstract

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 $\mathcal{Q}$ .