Determinantal and Pfaffian Hypersurfaces

The study of cubic surfaces in P3 is a classical subject. According to Meyer ([Me928]) it originated from the works of Plücker ([Pl829]) on intersection of quadrics and cubics and Magnus ([Ma833]) on maps of a plane by a linear system of cubics. However, it is a common thought that the theory of cubics surfaces started from Cayley’s and Salomon’s discovery of twenty-seven lines on a nonsingular cubic surface ([Ca849]). Salmon’s proof was based on his computation of the degree of the dual surface ([Sa847]), while Cayley’s proof used the count of tritangent planes through a line. In 1851 Sylvester claimed without a proof that a general cubic surface can be written uniquely as a sum of five cubes of linear forms ([Sy851]). This fact was proved ten years later by Clebsch ([Cl861]); we briefly mention it in section 2.5. In 1854 Schläfli discovered thirty-six double-sixers on a nonsingular cubic surface. In 1866 Clebsch proved that a general cubic surface can be obtained as the image of a birational map from the projective plane given by cubics through six points ([Cl866]). Using this he showed that the Schläfli’s notation ai; bj ; cij for twenty-seven lines correspond to the images of the exceptional curves, conics through five points and lines through two points.
This important result was independently proved by Cremona in his memoir ([Cr868]) of 1868. A modern treatment of the cubic surface — mainly on the complex field — as image of a birational map can be found especially in [Ha977], even though in terms of some advanced pieces of theory. We refer to [Ge989] for a quite easy construction of the cubic surface.
However, many mathematicians contributed to the understanding of smooth cubic surfaces: one can find very much in [He911], [Re988] about the geometry of the twenty-seven lines and in [HC952] about the geometry of Schläfli’s double-sixers.
The representation of codimension one curves and surfaces of small degree as linear determinants is a classical subject too. The case of cubic surfaces was already known in the middle of nineteenth century ([Gr855]); other examples of curves and surfaces are treated in [Sc881]. The general homogeneous forms which can be expressed as linear determinants are determined in [Di921], where Dickson showed that every curve has a determinantal representation. Determinantal representations in general were studied by many authors, for instance Beauville ([Be000]), Cook and Thomas ([CT979]), Room ([Ro938]). We refer especially to the first one for a modern development of determinantal representations of hypersurfaces.