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.
Rationality of cubic surfaces[edit]
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.[1] 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.[2] 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.[3]
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.[4] 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.
Real cubic surfaces[edit]
In contrast to the complex case, the space of smooth cubic surfaces over the real numbers is not connected in the classical topology (based on the topology of R). Its connected components (in other words, the classification of smooth real cubic surfaces up to isotopy) were determined by Ludwig Schläfli (1863), Felix Klein (1865), and H. G. Zeuthen (1875).[12] Namely, there are 5 isotopy classes of smooth real cubic surfaces X in , distinguished by the topology of the space of real points . The space of real points is diffeomorphic to either , or the disjoint union of and the 2-sphere, where denotes the connected sum of r copies of the real projective plane . Correspondingly, the number of real lines contained in X is 27, 15, 7, 3, or 3.
A smooth real cubic surface is rational over R if and only if its space of real points is connected, hence in the first four of the previous five cases.[13]
The average number of real lines on X is [14] when the defining polynomial for X is sampled at random from the Gaussian ensemble induced by the Bombieri inner product.
The moduli space of cubic surfaces[edit]
Two smooth cubic surfaces are isomorphic as algebraic varieties if and only if they are equivalent by some linear automorphism of . Geometric invariant theory gives a moduli space of cubic surfaces, with one point for each isomorphism class of smooth cubic surfaces. This moduli space has dimension 4. More precisely, it is an open subset of the weighted projective space P(12345), by Salmon and Clebsch (1860). In particular, it is a rational 4-fold.[15]
The cone of curves[edit]
The lines on a cubic surface X over an algebraically closed field can be described intrinsically, without reference to the embedding of X in : they are exactly the (−1)-curves on X, meaning the curves isomorphic to that have self-intersection −1. Also, the classes of lines in the Picard lattice of X (or equivalently the divisor class group) are exactly the elements u of Pic(X) such that and . (This uses that the restriction of the hyperplane line bundle O(1) on to X is the anticanonical line bundle , by the adjunction formula.)
For any projective variety X, the cone of curves means the convex cone spanned by all curves in X (in the real vector space of 1-cycles modulo numerical equivalence, or in the homology group if the base field is the complex numbers). For a cubic surface, the cone of curves is spanned by the 27 lines.[16] In particular, it is a rational polyhedral cone in with a large symmetry group, the Weyl group of . There is a similar description of the cone of curves for any del Pezzo surface.
Cubic surfaces over a field[edit]
A smooth cubic surface X over a field k which is not algebraically closed need not be rational over k. As an extreme case, there are smooth cubic surfaces over the rational numbers Q (or the p-adic numbers ) with no rational points, in which case X is certainly not rational.[17] If X(k) is nonempty, then X is at least unirational over k, by Beniamino Segre and János Kollár.[18] For k infinite, unirationality implies that the set of k-rational points is Zariski dense in X.
The absolute Galois group of k permutes the 27 lines of X over the algebraic closure of k (through some subgroup of the Weyl group of ). If some orbit of this action consists of disjoint lines, then X is the blow-up of a "simpler" del Pezzo surface over k at a closed point. Otherwise, X has Picard number 1. (The Picard group of X is a subgroup of the geometric Picard group .) In the latter case, Segre showed that X is never rational. More strongly, Yuri Manin proved a birational rigidity statement: two smooth cubic surfaces with Picard number 1 over a perfect field k are birational if and only if they are isomorphic.[19] For example, these results give many cubic surfaces over Q that are unirational but not rational.
