Blowups of Dirac structures

Given a real, twisted Dirac structure L on a smooth manifold M, and a closed embedded submanifold N⊆ M of codimension $>1$, we characterise when L lifts to a smooth, twisted Dirac structure on the real projective blowup of M along N. This holds precisely when N is either a submanifold transverse to L (with no further restrictions) or a submanifold invariant for L, for which the Lie algebras transverse to N have all of the same constant height k≥ 0. We also classify Lie algebras satisfying this Lie-theoretic property. We recover a theorem of Polishchuk, which establishes that a Poisson structure lifts to a Poisson structure on the blowup of a submanifold exactly when the submanifold is invariant and the transverse Lie algebras have constant height $k=0$.