I'm writing a blues song. There are a few 2-bar sections in the verse where there are no vocal parts and I want the piano to improvise around a specific notated part (it's kind of, but not exactly, an answer phrase to the preceding vocal line). So, say, I've got a sequence of three chords and I want to say "in place of the second chord, you could also use this chord" or "before the third chord, you can add the following fill-in note" or "this lead-up can be replaced with a triplet" and so on. How do I do this, or is it assumed that the performer will improvise around the notated part without being explicitly told "improvise here using the following variations"?
Good improvisers will not play over another soloist or vocalist, but if they can see, on their parts, that there's a gap to fill, they'll probably go for it naturally, just like a drummer waits till the end of a sung phrase, and puts a fill in while the vocalist takes another deep breath.
You could just write "ad lib" in that part. Perhaps with a suggestion of the harmony or rhythm but that may be obvious.
Either write what you want, or write 'fill'. The player WILL use appropriate material. If you're not sure of the player's ability to improvise, notate something appropriate but add 'or ad lib'.
Sounds to me like you should just write out what you want to hear. Improvising by definition is "in the moment composition". In the "old" days, I did a lot of studio work and the composers would either want to hear something specific (which is what I am hearing you say) or want to hear whatever I came up with.
If you give the player tips on what notes to play it essentially handcuffs the process of improvisation. I remember a session once where the writer wanted me to end a solo on a particular note. I would get into the solo and invariably screw up at the end because the realization of ending on a particular note was like being waken up from a dream or "taken out of the moment" so to speak.
If you have a good idea of what you want, why not just write it out. Nothing wrong with doing it that way.