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On Open Borel Orbits in Conormal Bundles to Schubert Cells
Abstract
Let C(v) denote a Schubert cell in the flag variety GLn/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(w0w) being contained in the smooth locus of X(w0v). 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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BibTeX
@misc{aoki2026open,
title = {{On Open Borel Orbits in Conormal Bundles to Schubert Cells}},
author = {Derek R. Aoki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35923},
eprint = {2610.35923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35923},
}
RIS
TY - UNPB
TI - On Open Borel Orbits in Conormal Bundles to Schubert Cells
AU - Derek R. Aoki
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.35923
UR - https://arxiv.org/abs/2610.35923
AB - Let C(v) denote a Schubert cell in the flag variety 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.
ER -