Title: 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 )} ⊆ 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 ⊆ 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.