  For curved projective manifolds we introduce a notion of a normal tractor
frame field, based around any point. This leads to canonical systems of
(redundant) coordinates that generalise the usual homogeneous coordinates on
projective space. These give preferred local maps to the model projective space
that encode geometric contact with the model to a level that is optimal, in a
suitable sense. In terms of the trivialisations arising from the special
frames, normal solutions of classes of natural linear PDE (so-called first BGG
equations) are shown to be necessarily polynomial in the generalised
homogeneous coordinates; the polynomial system is the pull back of a polynomial
system that solves the corresponding problem on the model. Thus questions
concerning the zero locus of solutions, as well as related finer geometric and
smooth data, are reduced to a study of the corresponding polynomial systems and
algebraic sets. We show that a normal solution determines a canonical manifold
stratification that reflects an orbit decomposition of the model. Applications
include the construction of structures that are analogues of Poincare-Einstein
manifolds.
