Classifying Smooth Quot Schemes

Roy Skjelnes; Gregory G. Smith; Michael Stillman

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arXiv · 2608.14424

Classifying Smooth Quot Schemes

Roy Skjelnes·Gregory G. Smith·Michael Stillman

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Abstract

The Quot scheme $\operatorname{Quot}^{q}(\mathcal{O}_{\mathbb{P}^n}^{r})$ parametrizes the quotients of the trivial vector bundle of rank $r$ on $n$-dimensional projective space that have Hilbert polynomial $q$ and are flat over a base scheme. We identify numerical conditions on the polynomial $q$ that completely determine when this Quot scheme is smooth and irreducible. Our approach also uncovers further geometric features of the projective scheme $\operatorname{Quot}^{q} (\mathcal{O}_{\mathbb{P}^n}^{r})$ including the smoothness of the lexicographic point.

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Roy Skjelnes, Gregory G. Smith, Michael Stillman. 2026-08-14. Classifying Smooth Quot Schemes. https://arxiv.org/abs/2608.14424

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