Representation stability for the Kontsevich space of stable maps

TL;DR

This paper proves a representation stability theorem for the homology of Kontsevich spaces of stable maps to a fixed algebraic variety, describing how the symmetric group representations stabilize as the number of marked points grows.

Contribution

It introduces a novel application of the category of finite sets and surjections to establish stability in the homology representations of Kontsevich spaces.

Findings

Representation stability holds for the homology of Kontsevich spaces

The stability is governed by the category of finite sets and surjections

Results apply for sufficiently large number of marked points

Abstract

For a fixed algebraic variety $X$, curve class $\alpha \in N_1(X)$, and genus $g \in \mathbb N$, we consider the sequence of $S_n$ representations obtained from the homology of the Kontsevich space of stable maps to $X$, $\bar M_{g,n}(X,\alpha)$. Using the category of finite sets and surjections, we prove a representation stability theorem that governs the behavior of this sequence of representations for $n$ sufficiently large.