Curves on K3 surfaces

Abstract

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces.(1) Regeneration, a process opposite to specialisation, which preserves the geometric genus and does not require the class of the curve to extend. (2) The marked point trick, which allows a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic zero. CONTENTSConjecture 1.1. Let X be a projective K3 surface over an algebraically closed field. Then X contains infinitely many rational curves. This conjecture has been established in many important cases: for very general K3 surfaces in characteristic zero [Che99, CL13], for K3 surfaces with odd Picard rank [LL11] (building on ideas of [BHT11]), for elliptic K3 surfaces and K3 surfaces with infinite automorphism groups [BT00, Tay18], and for some special K3

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

“…Otherwise, there is one specialization of residual charactertic different from 2 and 3 and which is supersingular, hence admits an elliptic firation by the Tate conjecture. By [Tay18b], such specialization admits geometrically infinitely many rational curves and we conclude by Proposition 5.1 in [CGL19]. 8.2.…”

Deformation of pairs and semiregularity

“…It is known that when X 0 is a K3 surface and the image ϕ 0 (C 0 ) is reduced, then the map ϕ 0 deforms to general fibers if the class [ϕ 0 (C 0 )] remains Hodge. This claim is proved using the twistor family associated with the hyperkähler structure of K3 surfaces, see for example [3]. Corollary 18 gives a generalization of this fact to general surfaces.…”

Curves of maximal moduli on K3 surfaces

“…Proof. The first claim is an application of the usual Arbarello-Cornalba Lemma in the case of K3 surfaces (see, e.g., [DS17]), whereas the second and third follow essentially from the first (see [CGL19,§2] and the proof of [CGL19, Lemma 6.3]).…”

“…Following the work of many people, it was recently proved in [CGL19] that for any integer g ≥ 0 and any complex K3 surface X, there is a sequence of integral curves C n ⊂ X of geometric genus g ≥ 0 such that for any ample divisor H lim n→∞ HC n = ∞.…”