F4 and desmic quartic surfaces

TL;DR

This paper explores the desmic pencil of quartic surfaces, revealing its connection to the Weyl group of F4 and Lie algebraic structures, highlighting its unique properties in algebraic surface theory.

Contribution

It uncovers the relationship between the desmic quartic surfaces and the Weyl group of F4, integrating classical algebraic geometry with Lie theory.

Findings

Desmic quartic surfaces are linked to the Weyl group of F4.

The desmic pencil can be derived from symmetric spaces related to Lie algebras.

Main properties are analyzed through a Lie theoretic framework.

Abstract

The desmic pencil of quartic surfaces is part of a beautiful, but mostly forgotten chapter of the classical theory of algebraic surfaces: it is the only non-degenerate pencil of surfaces in P3 containing at least three completely reducible members. We observe in this note that it is closely related to the Weyl group of the root system F4, and can be recovered from a series of symmetric spaces deduced from the exceptional Lie algebras. We discuss the main properties of the pencil from this Lie theoretic point of view.