Geodesic
Curves of maximal moduli on K3 surfaces
Forum of Mathematics, Sigma, Tome 10 (2022)

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We prove that if X is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus g on X with maximal, i.e., g-dimensional, variation in moduli. In particular, every K3 surface contains a curve of geometric genus 1 which moves in a nonisotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms. 
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     author = {Xi Chen and Frank Gounelas},
     title = {Curves of maximal moduli on {K3} surfaces},
     journal = {Forum of Mathematics, Sigma},
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Xi Chen; Frank Gounelas. Curves of maximal moduli on K3 surfaces. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.24

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