We study the interplay of the moduli of curves and the moduli of 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 surface. The decomposition has terms supported on Noether–Lefschetz loci which are not visible in the Beauville–Voisin decomposition for a fixed 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 surfaces is generated in Chow by the classes of the Noether–Lefschetz loci. Explicit boundary relations are constructed for all 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 surfaces. The WDVV relation in genus 0 is used in our proof of the MOP conjecture.
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Rahul Pandharipande, Qizheng Yin, Relations in the tautological ring of the moduli space of surfaces. J. Eur. Math. Soc. 22 (2020), no. 1, pp. 213–252DOI 10.4171/JEMS/920