The motivic class of the space of genus $0$ maps to the flag variety

Abstract

Let $\mathbb{k}$ be a field, let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}_{\mathbb{k}}$, 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 $K_{0}(\operatorname{Var}_{\mathbb{k}} )$, the Grothendieck group of varieties, is given by \[ [Ω^{2}_β(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{D-n^2}], \] where $D=\operatorname{dim}\bigl(Ω^{2}_β(\operatorname{Fl}_{n+1})\bigr)$. 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).