Hyperelliptic $d$-osculating covers and rational surfaces

TL;DR

This paper studies hyperelliptic covers of elliptic curves with a focus on their osculating order and associated rational surfaces, providing new characterizations and constructions of such covers and rational curves.

Contribution

It introduces the concept of osculating order for hyperelliptic covers and constructs rational curves on associated anticanonical rational surfaces.

Findings

Characterization of hyperelliptic covers by osculating order.

Construction methods for rational curves on anticanonical rational surfaces.

Identification of maximal genus covers with given osculating order.

Abstract

Let $\mathbb{P}^1$ and $(X,q)$ denote, respectively, the projective line and a fixed elliptic curve marked at its origin, both defined over an algebraically closed field $\mathbb{K}$ of arbitrary characteristic $\emph{\textbf{p}} \neq2$. We will consider all finite separable marked morphisms $\pi :(\Gamma ,p)\rightarrow (X,q)$, such that $\Gamma$ is a degree-$2$ cover of $ \mathbb{P}^1$, ramified at the smooth point $p \in \Gamma$. Canonically associated to $\pi$ there is the Abel (rational) embedding of $\Gamma$ into its \emph{generalized Jacobian}, $A_p: \Gamma \to Jac\,\Gamma$, and $\{0\} \subsetneq V^1_{\Gamma,p}...\subsetneq V ^g_{\Gamma,p}$, the flag of hyperosculating planes to $A_p(\Gamma)$ at $A_p(p)\in Jac\,\Gamma$ (cf. \textbf{2.1. & 2.2.}). On the other hand, we also have the homomorphism $\iota_\pi: X \to \Jac\,\Gamma$, obtained by dualizing $\pi$. There is a smallest positive integer $d$ such that the tangent line to $\iota_\pi( X)$ is contained in $V^d_{\Gamma,p}$. We call it \emph{the osculating order} of $\pi$. Studying, characterizing and constructing those with given \textit{osculating order} $d$ but maximal possible arithmetic genus, is one of the main issues. The other one, to which the first issue reduces, is the construction of all rational curves in a particular anticanonical rational surface associated to $X$ (i.e.: a rational surface with an effective anticanonical divisor).