On the supersingular reduction of K3 surfaces with complex multiplication

Ito, Kazuhiro

doi:10.48550/arxiv.1710.00225

Cited by 1 publication

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“…Let X be a K3 surface with complex multiplication. Then, X is defined over Q and there exists a prime ideal p such that X has good and supersingular reduction at p by [Ito17]. Thus, Proposition 3.9 and Proposition 5.1 show that X contains infinitely many rational curves.…”

Section: Applications Of the Regeneration Techniquementioning

confidence: 94%

Curves on K3 surfaces

Chen¹,

Gounelas²,

Liedtke³

2019

Preprint

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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

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Section: Applications Of the Regeneration Techniquementioning

confidence: 94%