Effective Serre Vanishing

Question

Suppose that $X = \mathbb{P}^n_k$ and $G$ is a coherent sheaf on $X$.

Question: Is there a way to determine some integer $n_0$ such that $H^1(X, G \otimes O_X(n)) = 0$ for all $n \geq n_0$? (obviously $n_0$ depends on $G$)

Of course, the higher cohomologies would be very interesting as well, but perhaps the above is simpler. One can also ask this for the relative case as well $f : Y \to X$ a projective morphism.

What I'd be particularly interested in is something that can be implemented into a computer.

In my particular case, I know I only have to compute some other things until Serre vanishing hits, at that point, I know I can stop computing. Unfortunately, I don't know how to tell when I've arrived. You can assume I have a presentation of my module $G$ if it helps.

PS: Actually, what I really want is vanishing of $$R^1 f_* (G(-nE))$$ for a blow-up $f : Y \to X$ with $E$ the pullback of the blownup ideal.

Answer 1

Take $n_0$ as (Castelnuovo–Mumford) regularity of $G$ minus $1$. This generalizes to higher cohomology: for $H^i$ take the regularity minus $i$

Let $M$ be a module representing $G$. Then, the (Castelnuovo–Mumford) regularity of $M$ is an upper bound of the regularity of $G$. So for an implementation you can take the regularity of $M$, which can be computed using a free resolution of $M$.

If you need a better bound then you can compute a "better" module for $G$ by approximating its module of global sections. However, this is rather expensive to my experience.