Vector bundles and the icosahedron

TL;DR

This paper explores the geometry of harmonic cubics, vector bundles, and the icosahedral group, linking classical algebraic geometry with modern vector bundle theory.

Contribution

It applies Atiyah's classification to harmonic cubics, connecting vector bundles, the Mukai-Umemura threefold, and the icosahedral symmetry in a novel geometric framework.

Findings

Classification of invariant strata in harmonic cubics

Identification of the Mukai-Umemura threefold within this context

Connections between vector bundles, cubic surfaces, and icosahedral symmetry

Abstract

A plane curve C defined by a homogeneous polynomial satisfying Laplace's equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two vector bundle defined on C. We consider the case of degree 3, and apply Atiyah's classification of bundles to determine various invariant strata in the space of harmonic cubics. We encounter the Mukai-Umemura threefold and link up with the classical geometry of the Clebsch diagonal cubic surface, certain distinguished rational curves on it, and the action of the icosahedral group.