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. 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 a given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness of the complement using $\mathbb{A}^1$-curves parameterized by each irreducible component. I'm going to explain how the key to these results are the strangeness of all $\mathbb{A}^1$-curves.

algebraic geometry

Audience: researchers in the topic

Algebraic Geometry NorthEastern Series (AGNES)