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Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves

Jonas Bergström, Thomas Wennink

Abstract

We have written a computer program that implements Deligne's pullback and pushforward weight spectral sequences to compute the weight graded pieces of the rational cohomology of moduli spaces of pointed smooth curves (as well as curves of compact type and curves with rational tails) in cases where the cohomology groups appearing in the boundary stratification of the Deligne-Mumford compactification are generated by tautological classes (and when Pixton's relations are all relations). The weight graded pieces are computed together with the induced action of the symmetric group permuting the points on the curves. Using the computer program we have determined this information in the case of genus five as well as in the case of genus three with three marked points.

Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves

Abstract

We have written a computer program that implements Deligne's pullback and pushforward weight spectral sequences to compute the weight graded pieces of the rational cohomology of moduli spaces of pointed smooth curves (as well as curves of compact type and curves with rational tails) in cases where the cohomology groups appearing in the boundary stratification of the Deligne-Mumford compactification are generated by tautological classes (and when Pixton's relations are all relations). The weight graded pieces are computed together with the induced action of the symmetric group permuting the points on the curves. Using the computer program we have determined this information in the case of genus five as well as in the case of genus three with three marked points.

Paper Structure

This paper contains 12 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Deligne's pushforward weight spectral sequence
  3. Compact type and rational tails
  4. Deligne's pullback weight spectral sequence
  5. Compact type and rational tails
  6. Tautological cohomology
  7. The computer implementation
  8. The algorithm
  9. Computing the tautological ring of the moduli space of stable curves
  10. Bookkeeping challenges
  11. Computations
  12. Further applications

Key Result

Theorem 1.1

The cohomology of $\mathcal{M}_{5}$ is of Tate type and equals,

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Example 2.1
  • Example 3.1
  • Remark 5.1