Strangeness of A1-curves

Abstract

A (plane) curve is called strange if all of its tangent lines pass through a fixed point, called the strange point. We study the $\mathbb{A}^1$-geometry, and moduli spaces of $\mathbb{A}^1$-curves on the complement of a rational strange plane curve $\Delta$ of degree $p$. On the one hand we observe that $\PP^2\setminus \Delta $ is $\mathbb{A}^1$-connected. On the other hand, these $\mathbb{A}^1$-curves can have large obstructions: therefore $\\mathbb{P}^2\setminus \Delta$ is only inseparably $\mathbb{A}^1$-connected. To understand the strange properties of $\mathbb{A}^1$-curves we study the moduli spaces which parameterize them. In each characteristic, we show the moduli spaces of $\mathbb{A}^1$-curves are connected and classify their irreducible components. The key to the above results are deformation of $\mathbb{A}^1$-curves along certain ``logarithmic" foliations. A direct, and very surprising, consequence of the existence of this foliation is that all $\mathbb{A}^1$-curves are strange. Using the above geometry, we construct for each $p$, new and very explicit examples of supercuspidal families of curves. That is, a fibration with smooth total space but {\it every} fiber is singular with a large number of cusps.