Nonlinearizable embeddings of elliptic curves in rational surfaces

TL;DR

This paper demonstrates that for any smooth cubic in the projective plane, there are configurations of nine points where the resulting blown-up surface contains a non-linearizable elliptic curve with a nontorsion normal bundle, addressing a 1975 problem.

Contribution

It constructs explicit configurations of points on cubic curves in the plane where the associated elliptic curves are non-linearizable in rational surfaces, solving a longstanding open problem.

Findings

Existence of dense configurations with non-linearizable elliptic curves

Construction of elliptic curves with nontorsion normal bundles

Resolution of Ogus's 1975 problem

Abstract

We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_\delta$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.