Cozyleo

We first prove a lemma on generic flatness: LEMMA (Generic Flatness) Let $A$ be a noetherian domain, and $B$ a finite type $A$-algebra. Let $M$ be a finite $B$-module. Then there exists an $f \in A, f\neq 0$, such that the localization $M_f$ is a free module over $A_f$. \begin{proof} Let $K$ be the fraction… — read more

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… — read more