Relations in the tautological ring of the moduli space of $K3$ surfaces

Abstract

We study the interplay of the moduli of curves and the moduli of $K3$ surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler’s relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal $K3$ surface. The decomposition has terms supported on Noether–Lefschetz loci which are not visible in the Beauville–Voisin decomposition for a fixed $K3$ surface. As a result of our universal decomposition, we prove the conjecture of Marian–Oprea–Pandharipande: the full tautological ring of the moduli space of $K3$ surfaces is generated in Chow by the classes of the Noether–Lefschetz loci. Explicit boundary relations are constructed for all $\kappa$ classes. More generally, we propose a connection between relations in the tautological ring of the moduli spaces of curves and relations in the tautological ring of the moduli space of $K3$ surfaces. The WDVV relation in genus 0 is used in our proof of the MOP conjecture.