The motivic class of the space of genus $0$ maps to the flag variety
Jim Bryan, Balázs Elek, Freddie Manners, George Salafatinos, Ravi Vakil
TL;DR
This paper studies the space Ω^{2}_β(Fl_{n+1}) of based genus-zero maps to the complete flag variety, focusing on the case where β is strictly monotonic. It proves a motivic formula in the Grothendieck ring: [Ω^{2}_{d_n,...,d_1}(Fl_{n+1})] = [GL_n × 𝔸^{D−n^2}] with D = ∑_{k=1}^n 2d_k, under mild positivity assumptions, and deduces implications for finite-field point counts and connections to the topology of double loop spaces. The proof proceeds via a tower of partial flag varieties, showing each step is a motivically trivial fibration and computing fiber classes through N_p(E) constructions tied to Grothendieck ring computations; a detailed stratification by splitting types and depth is used to control the geometry. The paper also discusses the limits of such motivic results by presenting negative examples where motivic equality does not imply isomorphism types or rational homotopy equivalences, highlighting the delicate relationship between motivic and topological invariants in this setting.
Abstract
Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $Ω^{2}_β(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=β$. We show that under a mild positivity condition on $β$, the class of $Ω^{2}_β(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [Ω^{2}_β(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).