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Zero cycles on the moduli space of curves

Rahul Pandharipande, Johannes Schmitt

Abstract

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

Zero cycles on the moduli space of curves

Abstract

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

Paper Structure

This paper contains 25 sections, 13 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Moduli of curves
  3. $0$-cycles in the tautological ring
  4. Tautological points
  5. Curves on surfaces
  6. Further results on tautological $0$-cycles
  7. $T$-numbers for $K3$ surfaces
  8. Plan of the paper
  9. Acknowledgements
  10. Basic results about cycles and curves
  11. Rational surfaces
  12. Proof of Theorem \ref{['curveratsur']}
  13. Variations
  14. K3 surfaces
  15. Beauville-Voisin classes
  16. ...and 10 more sections

Key Result

Proposition \oldthetheorem

For all $(g,n)$, we have $R_0({\overline{\mathcal{M}}}_{g,n}) \stackrel{\sim}{=} \mathbb{Q}$.

Figures (1)

  • Figure 1: $\overline{\mathcal{M}}_{g,n}$ is rationally connected for $n \leq n_{\mathrm{max}}$, see benzobrunoverracasnatifontanarifarkaslogankodairaverra.

Theorems & Definitions (16)

  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • ...and 6 more