Integral sections of elliptic surfaces and degenerated (2,3) torus decompositions of a 3-cuspidal quartic

Abstract

In this note, we consider when a plane curve given by a polynomial of the form

$$x^3+a_1\left(t\right)x^2+a_2\left(t\right)x+a_3\left(t\right)=0,$$

where ${\text{deg}}_t{a}_i\left(t\right)\le id\left(d\mathrm{:\hspace{0.17em}}\text{even}\right),$ has degenerated $(2,3)$ torus decompositions by using arithmetic properties of elliptic surfaces and show that a 3-cuspidal quartic has infinitely many degenerated $(2,3)$ torus decompositions.