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 (e.g., Ivan Smith's Floer cohomology and pencils of quadrics provides some evidence in that direction). Now, such an equivalence should probably have some geometric meaning via SYZ perhaps and be explained in terms of a nice mirror recipe.