The projective characterization of elliptic plane curves which have one place at infinity

TL;DR

This paper characterizes smooth affine elliptic plane curves with one place at infinity by their cusp properties and self-intersection numbers, establishing a maximality result among similar curves.

Contribution

It provides a projective geometric characterization of elliptic curves with one cusp at infinity using self-intersection numbers.

Findings

Curves are identified with elliptic projective curves having a single cusp.

The self-intersection number of the strict transform is characterized and maximized.

The maximal self-intersection number among such elliptic curves is established.

Abstract

In this paper we consider smooth affine elliptic plane curves having one place at infinity. We identify them with elliptic projective plane curves having only one cusp as their singular points and meeting with the line at infinity only at the cusp. We characterize such curves by the self-intersection number of the strict transform of them via the minimal embedded resolution of their cusp. Furthermore, we prove that the self-intersection number of them is the maximum value among those of all the elliptic plane curves having only one cusp.