Question: Let's assume we have a pair $(X,\check{X})$ of Calabi-Yau n-folds which is expected to be mirror dual to each other. Now take a subvariety $Y \subset X$ and consider $Z = Bl_Y X$ (Really, $Z$ is probably Fano/general type, but for starters, we can also assume that it is again a CY n-fold).

• Are there any examples where such mirror pairs is known?
• Ideally: is there a general recipe for what should be the mirror $\check{Z}$?

Motivation: Derived categories behave nicely under blowup; In fact by Bondal-Orlov, we can construct the derived category of sheaves on $Z$ from that of $X$ and $Y$. Thus one should expect the same for Fukaya categories. Now, such an equivalence should probably have some geometric meaning via SYZ perhaps and be explained in terms of a nice mirror recipe.