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, prove almost all cases in positive characteristic 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. Regeneration, a process opposite to specialisation, which preserves the geometric genus and does not require the class of the curve to extend, and 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.
Cite
@article{arxiv.1907.01207,
title = {Curves on K3 surfaces},
author = {Xi Chen and Frank Gounelas and Christian Liedtke},
journal= {arXiv preprint arXiv:1907.01207},
year = {2023}
}
Comments
65 pages. Various corrections and new appendix producing immersed rational curves from a smooth degeneration. Final version, to appear in Duke Math Journal