Families of algebraic and continuous maps to $\mathbb{P}^m$
Alexis Aumonier
TL;DR
The paper develops a parameterized Segal-type comparison between spaces of algebraic and continuous maps to $\mathbb P^m$ in families over a base stack. It introduces a fibrewise degree class in $H^0(\mathfrak B^{top}; R^2\pi_*\underline{\mathbb Z})$ and defines algebraic and topological moduli stacks $\mathrm{Alg}_{\mathfrak B}^\alpha(\mathfrak X, \mathbb P^m)$ and $\mathrm{Map}_{\mathfrak B}^\alpha(\mathfrak X, \mathbb P^m)$, proving that the natural map between them induces a singular homology isomorphism in a range governed by the uniform ampleness bound $d(\mathfrak X,\alpha)$. The method reduces to the scheme level using étale presentations and a Poincaré bundle, then compares semi-simplicial models to pass to the stack level. As a key corollary, the authors recover and extend rational-homology stability results for families of maps from curves, including explicit ranges when the base is the moduli stack of curves $\mathfrak M_g$, and provide a direct analytic-combinatorial proof in that setting. The work connects to Ayala's results on moduli of curves with maps and demonstrates how homotopical tools can compute homology of intricate moduli problems. The findings illuminate the interplay between algebraic and topological mapping data in families, with potential implications for understanding global geometry of moduli spaces.
Abstract
We explain how results comparing the homology of spaces of algebraic and continuous maps to projective spaces can be leveraged to compare moduli stacks of families of algebraic and continuous maps.