Coskun, Emre; Kulkarni, Rajesh S.; Mustopa, Yusuf

Pfaffian quartic surfaces and representations of Clifford algebras

Doc. Math., J. DMV 17, 1003-1028 (2012)


Summary: 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.

Mathematics Subject Classification

14J60, 13C14, 16G30


ulrich bundles, algebraic surfaces, representations of Clifford algebras