Pfaffian quartic surfaces and representations of Clifford algebras

Abstract

Given a general 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 and van den Bergh's correspondence between representations of the generalized Clifford algebra $C_f$ associated to $f$ and Ulrich bundles on the surface $X_f:=w^4=f(x_1,x_2,x_3)\subseteq P^3$ to construct a positive-dimensional family of 8-dimensional irreducible representations of $C_f.$ The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result of Basili on complete intersection curves in $P^3$ to produce simple Ulrich bundles of rank 2 on a smooth quartic surface $X\subseteq P^3$ with determinant $O_X(3).$ This implies that every smooth quartic surface in $P^3$ is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.