  We study blow-ups in generalized complex geometry. To that end we introduce
the concept of holomorphic ideal, which allows one to define a blow-up in the
category of smooth manifolds. We then investigate which generalized complex
submanifolds are suitable for blowing up. Two classes naturally appear;
generalized Poisson submanifolds and generalized Poisson transversals,
submanifolds which look complex, respectively symplectic in transverse
directions. We show that generalized Poisson submanifolds carry a canonical
holomorphic ideal and give a necessary and sufficient condition for the
corresponding blow-up to be generalized complex. For the generalized Poisson
transversals we give a normal form for a neighborhood of the submanifold, and
use that to define a generalized complex blow-up, which is up to deformation
independent of choices.
