Classifying Smooth Quot Schemes

The Quot scheme ${\mathrm{Quot}}^q({\mathcal{O}}_{{\mathbb{P}}^n}^r)$ 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 ${\mathrm{Quot}}^q({\mathcal{O}}_{{\mathbb{P}}^n}^r)$ including the smoothness of the lexicographic point.