The projective geometry of Freudenthal's magic square

TL;DR

This paper explores the geometric and algebraic structures underlying Freudenthal's magic square, providing unified descriptions of orbit closures and relating invariant polynomials to classical discriminants.

Contribution

It offers a novel geometric framework connecting algebraic geometry, representation theory, and composition algebras within Freudenthal's magic square.

Findings

Unified geometric descriptions of orbit closures

Interpretation of invariant quartic polynomials as generalized discriminants

Descriptions of hyperplane sections and desingularizations

Abstract

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.