  Little is known about the behaviour of the Oka property of a complex manifold
with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$
if it is the complement of an algebraic subvariety of codimension at least $2$
in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A
manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and
a manifold of Class $\mathscr A$ blown up at finitely many points is of Class
$\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up
along an arbitrary algebraic submanifold (not necessarily connected) is
algebraically subelliptic. For algebraic manifolds in general, we prove that
strong algebraic dominability, a weakening of algebraic subellipticity, is
preserved by an arbitrary blow-up with a smooth centre. We use the main result
to confirm a prediction of Forster's famous conjecture that every open Riemann
surface may be properly holomorphically embedded into $\mathbb C^2$.
