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

TL;DR

This paper computes the motivic class of the space of genus 0 maps to flag varieties, showing it equals a product of a general linear group and an affine space under certain conditions, using a novel collaborative approach.

Contribution

It provides a new explicit formula for the motivic class of genus 0 mapping spaces to flag varieties, combining human and AI-assisted methods.

Findings

Motivic class expressed as [GL_n × A^a]

Applicable under mild positivity conditions on the class β

Introduces a collaborative research approach with AI tools

Abstract

Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=\beta$. We show that under a mild positivity condition on $\beta$, the class of $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] 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).