Looking for Rational Curves on Cubic Hypersurfaces

The aim of these lectures is to study rational points and rational curves on varieties, mainly over finite fields Fq. We concentrate on hypersurfaces Xn of degree ≤ n+ 1 in Pn+1, especially on cubic hypersurfaces. The theorem of Chevalley–Warning (cf. Esnault’s lectures) guarantees rational points on low degree hypersurfaces over finite fields. That is, if X ⊂ Pn+1 is a hypersurface of degree ≤ n+ 1, then X(Fq) 6= ∅. In particular, every cubic hypersurface of dimension ≥ 2 defined over a finite field contains a rational point, but we would like to say more. • Which cubic hypersurfaces contain more than one rational point? • Which cubic hypersurfaces contain rational curves? • Which cubic hypersurfaces contain many rational curves? Note that there can be rational curves on X even if X has a unique Fqpoint. Indeed, f : P1 → X could map all q + 1 points of P(Fq) to the same point in X(Fq), even if f is not constant. So what does it mean for a variety to contain many rational curves? As an example, let us look at CP2. We know that through any 2 points there is a line, through any 5 points there is a conic, and so on. So we might say that a variety XK contains many rational curves if through any number of points p1, . . . , pn ∈ X(K) there is a rational curve defined over K. However, we are in trouble over finite fields. A smooth rational curve over Fq has only q + 1 points, so it can never pass through more than q + 1 points in X(Fq). Thus, for cubic hypersurfaces, the following result, proved in Section 9, appears to be optimal: Theorem 1.1. Let X ⊂ Pn+1 be a smooth cubic hypersurface over Fq. Assume that n ≥ 2 and q ≥ 8. Then every map of sets φ : P(Fq) → X(Fq) can be extended to a map of Fq-varieties Φ : P 1 → X.