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.