Chilean configuration of conics, lines and points

TL;DR

This paper constructs a family of geometric configurations involving conics and points in the projective plane using elliptic fibrations, revealing new connections to classical configurations like the Hesse configuration.

Contribution

It introduces a novel family of conic-point configurations in the projective plane linked to elliptic fibrations, expanding the understanding of geometric configurations.

Findings

Constructs a one-parameter family of configurations with 12 conics and 9 points

Shows the connection between these configurations and Halphen elliptic fibrations of index 2

Relates the configurations to classical geometric arrangements like the Hesse configuration

Abstract

Using the theory of rational elliptic fibrations, we construct and discuss a one parameter family of configurations of $12$ conics and $9$ points in the projective plane that realizes an abstract configuration $(12_6,9_8)$. This is analogous to the famous Hesse configuration of $12$ lines and $9$ points forming an abstract configuration $(12_3,9_4)$. We also show that any Halphen elliptic fibration of index $2$ with four triangular singular fibers arises from such configuration of conics.