The S_n-equivariant Euler characteristic of M_{1, n}(ℙ^r, d)

2024/12/16 by Kannan, Siddarth, Song, Terry Dekun
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2412.12317

Abstract

We compute the S_n-equivariant topological Euler characteristic of the Kontsevich moduli space M_{1, n}(ℙ^r, d). Letting M_{1, n}^nrt(ℙ^r, d) ⊂ M_{1, n}(¶^r, d) denote the subspace of maps from curves without rational tails, we solve for the motive of M_{1, n}(ℙ^r, d) in terms of M_{1, n}^nrt(ℙ^r, d) and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic ℂ^⋆-action on ℙ^r, we derive a closed formula for the Euler characteristic of M_{1, n}^nrt(ℙ^r, d)^{ℂ^⋆} as an S_n-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of M_1,n(ℙ^r, d). Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types A and B, as well as the enumeration of graph colourings with prescribed symmetry. We also prove a structural result about the S_n-equivariant Euler characteristic of M_{g, n}(ℙ^r, d) in arbitrary genus.

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