Trending in Algebraic Geometry
2601.07222The motivic class of the space of genus $0$ maps to the flag variety
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).
2601.07222
Jan 2026Algebraic Geometry
2512.01968Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds
We prove that if two very general cubic fourfolds are L-equivalent then they are isomorphic, and we observe that there exist special cubic fourfolds which are L-equivalent but not isomorphic. As a consequence, we provide examples of hyper-Kähler manifolds which are L-equivalent but not birational. We also provide further examples in support of the fact that L-equivalent hyper-Kähler manifolds should be D-equivalent, as conjectured by Meinsma.
2512.019681
Dec 2025Algebraic Geometry
