2 - Cubic surfaces

Summary

In Chapter 1, we produced many examples of rational cubic surfaces. We also discussed the fact that a smooth cubic surface is unirational if and only if it has a rational point. In this chapter, we treat the subtle issue of rationality for smooth cubic surfaces more systematically. In particular, we show that there are many cubic surfaces defined over ℚ which are not rational over ℚ.

We begin in Section one by stating our main results, which provide a complete understanding of rationality issues for smooth cubic surfaces of Picard number one. These results are Segre's theorem, stating that such a surface is never rational, and the related result of Manin stating that any two such birationally equivalent surfaces are actually isomorphic. In the second section, we set up the general machinery of linear systems to study birational maps of surfaces. This technique, called the Noether–Fano method, is quite powerful and ultimately leads to a proof that smooth quartic threefolds are not rational, in Chapter 5. In this chapter, however, we apply this method only to the case of cubic surfaces, proving the theorems of Segre and Manin in Section 3.

Over an algebraically closed field, every cubic surface has Picard number seven, and it is not obvious that there is any cubic surface with Picard number one. Thus, in Section 4, we develop criteria for checking whether a cubic surface has Picard number one.