On the Chow and cohomology rings of moduli spaces of stable curves

2022/08/03 by Canning, Samir, Larson, Hannah · 2 citations
#14C15 #14C17 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2208.02357

Abstract

In this paper, we ask: for which (g, n) is the rational Chow or cohomology ring of M_g,n generated by tautological classes? This question has been fully answered in genus 0 by Keel (the Chow and cohomology rings are tautological for all n) and genus 1 by Belorousski (the rings are tautological if and only if n ≤ 10). For g ≥ 2, work of van Zelm shows the Chow and cohomology rings are not tautological once 2g + n ≥ 24, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of M_g,n are isomorphic and generated by tautological classes for g = 2 and n ≤ 9 and for 3 ≤ g ≤ 7 and 2g + n ≤ 14. For such (g, n), this implies that the tautological ring is Gorenstein and M_g,n has polynomial point count.

Cited by

• On the Chow Rings of the Moduli Spaces M_5,8 and M_5,9
• The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs II

Related

• The Chow ring of the moduli space of curves of genus 6
• Tautological and non-tautological cohomology of the moduli space of curves
• Tautological rings of stable map spaces
• Chow rings of stacks of prestable curves II
• Logarithmic tautological rings of the moduli spaces of curves