Cubic surface

A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3 (hence, cubic). Cubic surfaces are del Pezzo surfaces. A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3 (hence, cubic). Cubic surfaces are del Pezzo surfaces. If P 3 {displaystyle mathbb {P} ^{3}} has homogeneous coordinates [ X : Y : Z : W ] {displaystyle } , then the set of points where is a cubic surface called the Fermat cubic surface. The Clebsch surface is the set of points where Cayley's nodal cubic surface is the set of points where The Cayley–Salmon theorem (Cayley 1849) states that a smooth cubic surface over an algebraically closed field contains 27 straight lines. These can be characterized independently of the embedding into projective space as the rational lines with self-intersection number −1, or in other words the −1-curves on the surface. An Eckardt point is a point where 3 of the 27 lines meet. A smooth cubic surface can also be described as a rational surface obtained by blowing up six points in the projective plane in general position (in this case, “general position” means no three points are aligned and no six are on a conic section). The 27 lines are the exceptional divisors above the 6 blown up points, the proper transforms of the 15 lines in P 2 {displaystyle mathbb {P} ^{2}} which join two of the blown up points, and the proper transforms of the 6 conics in P 2 {displaystyle mathbb {P} ^{2}} which contain all but one of the blown up points. Clebsch gave a model of a cubic surface, called the Clebsch diagonal surface, where all the 27 lines are defined over the field Q, and in particular are all real. The 27 lines can also be identified with some objects arising in representation theory. In particular, these 27 lines can be identified with 27 vectors in the dual of the E6 lattice so their configuration is acted on by the Weyl group of E6. In particular they form a basis of the 27-dimensional fundamental representation of the group E6.