Geodesic
Pfaffian quartic surfaces and representations of Clifford algebras
Documenta mathematica, Tome 17 (2012), pp. 1003-1028
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Given a general ternary form f=f(x1​,x2​,x3​) 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 Cf​ associated to f and Ulrich bundles on the surface Xf​:=w4=f(x1​,x2​,x3​)⊆P3 to construct a positive-dimensional family of 8-dimensional irreducible representations of Cf​. 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 P3 to produce simple Ulrich bundles of rank 2 on a smooth quartic surface X⊆P3 with determinant OX​(3). This implies that every smooth quartic surface in P3 is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.
DOI : 10.4171/dm/388
Classification : 13C14, 14J60, 16G30
Mots-clés : algebraic surfaces, ulrich bundles, representations of Clifford algebras 
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     author = {Emre Coskun and Rajesh S. Kulkarni and Yusuf Mustopa},
     title = {Pfaffian quartic surfaces and representations of {Clifford} algebras},
     journal = {Documenta mathematica},
     pages = {1003--1028},
     year = {2012},
     volume = {17},
     doi = {10.4171/dm/388},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/388/}
}
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Emre Coskun; Rajesh S. Kulkarni; Yusuf Mustopa. Pfaffian quartic surfaces and representations of Clifford algebras. Documenta mathematica, Tome 17 (2012), pp. 1003-1028. doi: 10.4171/dm/388

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