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arXiv · 2610.35923

On Open Borel Orbits in Conormal Bundles to Schubert Cells

Derek R. Aoki
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Abstract

Let $C(v)$ denote a Schubert cell in the flag variety $\operatorname{GL}_n/B$, and let $X(w)$ be a Schubert variety containing $C(v)$. Mellit has conjectured that, if $X(w)$ is smooth, the existence of an open $B$-orbit in the conormal bundle to $C(v)$ in $X(w)$ is equivalent to $C(w_0w)$ being contained in the smooth locus of $X(w_0v)$. We prove a stronger version of this conjecture, requiring only that $X(w)$ is factorial. We prove our result by constructing a sequence of graphs associated to the pair $v,w$ such that the existence of a cycle in one of these graphs is equivalent to the non-existence of such an open orbit.

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Derek R. Aoki. 2026-10-13. On Open Borel Orbits in Conormal Bundles to Schubert Cells. https://arxiv.org/abs/2610.35923

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