Cubic surface

In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space . The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface in . Many properties of cubic surfaces hold more generally for del Pezzo surfaces. A central feature of smooth cubic surfaces X over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane minus a lower-dimensional subset and X minus a lower-dimensional subset. More generally, every irreducible cubic surface (possibly singular) over an algebraically closed field is rational unless it is the projective cone over a cubic curve. In this respect, cubic surfaces are much simpler than smooth surfaces of degree at least 4 in , which are never rational. In characteristic zero, smooth surfaces of degree at least 4 in are not even uniruled. More strongly, Clebsch showed that every smooth cubic surface in over an algebraically closed field is isomorphic to the blow-up of at 6 points. As a result, every smooth cubic surface over the complex numbers is diffeomorphic to the connected sum , where the minus sign refers to a change of orientation. Conversely, the blow-up of at 6 points is isomorphic to a cubic surface if and only if the points are in general position, meaning that no three points lie on a line and all 6 do not lie on a conic. As a complex manifold (or an algebraic variety), the surface depends on the arrangement of those 6 points. Most proofs of rationality for cubic surfaces start by finding a line on the surface.