Pfaffian quartic surfaces and representations of Clifford algebras
By Emre Coskun, Rajesh S. Kulkarni, Yusuf Mustopa. Documenta Mathematica, 2011
56 citations recorded by the source corpus.
Given a nondegenerate 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 $X_f:=\{w^{4}=f(x_1,x_2,x_3)\} \subseteq \mathbb{P}^3$ to construct a positive-dimensional family of 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 $\mathbb{P}^{3}$ to produce simple Ulrich bundles of rank 2 on a smooth quartic surface $X \subseteq \mathbb{P}^3$ with determinant $\mathcal{O}_X(3).$ This implies that every smooth quartic surface in $\mathbb{P}^3$ is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0e9d217c68b286b6f8b90a80af707a8ec033b378
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Citations in the corpus
- n-dimensional Fano varieties of wild representation type (2010, background): “In [1] (extending results from [7]) the moduli space of stable Ulrich bundles on a polarized K3 surface X ⊂ P is thoroughly studied.”