# Pfaffian quartic surfaces and representations of Clifford algebras

### Emre Coskun

Department of Mathematics Department of Mathematics Middle East Technical Uni- Michigan State University versity East Lansing Ankara MI 48824 TURKEY 06800### Rajesh S. Kulkarni

Department of Mathematics Department of Mathematics Middle East Technical Uni- Michigan State University versity East Lansing Ankara MI 48824 TURKEY 06800### Yusuf Mustopa

Department of Mathematics Boston College Chestnut Hill MA 02467

## 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.

## Cite this article

Emre Coskun, Rajesh S. Kulkarni, Yusuf Mustopa, Pfaffian quartic surfaces and representations of Clifford algebras. Doc. Math. 17 (2012), pp. 1003–1028

DOI 10.4171/DM/388