Cozyleo

The construction of Quot schemes mainly consists of two parts: Castelnuovo-Mumford regularity and Flattening stratification.

Let $k$ be a field and let $\mathcal{F}$ be a coherent sheaf on the projective space $\mathbb{P}^n$ over $k$. Let $m$ be an integer. The sheaf $\mathcal{F}$ is said to be $m$-regular if $$H^i(\mathbb{P}^n, \mathcal{F}(m-i)) = 0 \text{ for each } i \leq 1$$.