Plane ${\mathbb{A}}^1$-curves on the complement of strange rational curves

Abstract

A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study ${\mathbb{A}}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of ${\mathbb{A}}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable ${\mathbb{A}}^1$-connectedness via the ${\mathbb{A}}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all ${\mathbb{A}}^1$-curves. As an application, in every characteristic we construct explicit covering families of ${\mathbb{A}}^1$-curves, whose total spaces are smooth along large numbers of cusps on each general fiber.