Representations of Clifford algebras of ternary quartic forms

TL;DR

This paper explores the geometric construction of Clifford algebra representations for nondegenerate quartic forms in three variables, leading to new linear Pfaffian representations of associated quartic surfaces and insights into Brill-Noether theory.

Contribution

It introduces a geometric method using K3 surfaces to construct irreducible Clifford algebra representations for ternary quartic forms, revealing new Pfaffian representations and Brill-Noether properties.

Findings

Constructed a positive-dimensional family of irreducible representations.

Established the existence of linear Pfaffian representations of the quartic surface.

Provided information on the Brill-Noether theory of general smooth curves in a specific linear system.

Abstract

Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces to construct a certain positive-dimensional family of irreducible representations of the generalized Clifford algebra associated to $f.$ From this we obtain the existence of linear Pfaffian representations of the quartic surface $X_f=\{w^4=f(x_1,x_2,x_3)\},$ as well as information on the Brill-Noether theory of a general smooth curve in the linear system $|\mathcal{O}_{X_f}(3)|.$