**Introduction.** I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction of the "ultra"ness.

**An example.** Given a free (ultra)filter $F$ on $\omega$, **Grigorieff forcing** is defined as
$$ G(F) := \{ f:X \rightarrow 2: \omega \setminus X \in F \},$$
partially ordered by reverse inclusion. A simple density argument shows that **"$G(F)$ destroys $F$"**, i.e., the filter generated by $F$ in a generic extension is **not** an ultrafilter (the generic real being the culprit).

Of course, there are many forcing notions that specifically destroy ultrafilters (also, Bartoszynski, Judah and Shelah showed that whenever there's a new real in the extension, some ground model ultrafilter was destroyed).

My question is:

**If $F$ is destroyed, how far away is $F$ from being the ultrafilter it once was?**

Maybe a more positive version: **Which properties of $F$ can we destroy while preserving others?**

This might seem awfully vague, so before you vote to close let me explain what kind of answers I'm hoping for.

**Positive answers.**- If the forcing is $\omega^\omega$-bounding and $F$ is rapid, then $F$ will still be rapid. That's a very clean and simple preservation.
- In Shelah's model without P-points, all ground model Ramsey ultrafilters stop being P-points but "remain" Q-points.

**"Minimal" answers.**Is it possible that $F$ together with the generic real generates an ultrafilter, i.e., there are only two ultrafilters extending $F$? For Grigorieff forcing, I'd expect this needs at least a Ramsey ultrafilter. But maybe other forcings have this property?**Negative answers.**Say $F$ is a P-point; can $F$ still be extended to a P-point? Shelah tells us that forcing with the full product $G(F)^\omega$ denies this. Is it known whether $G(F)$ already denies this? Do other forcing notions allow this?

I know there is a lot of literature on **preserving ultrafilters** (mostly P-points, I think) but I'm more interested in the case where the ultrafilter is actually destroyed. But I'd welcome anything that sheds light on this.

PS: community wiki, of course.