I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in lecture 16 about which I feel confused.
Оказывается, что для пучков на проективном пространстве, полином Гильберта это единственный существенно дискретный инвариант: как только он зафиксирован, можно построить такое плоское семейство над конечномерной нётеровой базой $Y$, что любой пучок с данным $P(\mathcal{F},l)$ появляется в нем как слой, причем только один раз.
It can be translated as "It turns out that for sheaves on projective space Hilbert polynomial is the only essentially discrete invariant: once it is fixed, it is possible to construct a flat family over a finite-dimensional Noetherian base Y such that any sheaf with given $P(\mathcal{F}, l)$ appears in it as fiber exactly once".
I struggle to translate this remark into a precise mathematical statement. In particular, I don't understand what does 'exactly once' mean; if we have two points $y_1, y_2 \in Y$, how can we compare sheaves on $f^{-1}(y_1)$ and $f^{-1}(y_2)$? Can someone provide me the precise statement that Kaledin probably had in mind?
A "family of sheaves on $\mathbb{P}^n$ parametrized by $Y$" is defined as a sheaf on the product $\mathbb{P}^n \times Y$, so for every point $y \in Y$, we still get a sheaf on the same $\mathbb{P}^n$. To get a good notion, one needs to require that our sheaf on $\mathbb{P}^n \times Y$ is flat over $Y$ (but of course not necessarily over $\mathbb{P}^n \times Y$).
Update: If a morphism is flat, then roughly, it is flat if and only if the number of points in all the fibers is the same---but we have to count smartly---some fibers are not smooth, and then we have to count with multiplicities.