F4 and desmic quartic surfaces

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.