Generic geometry, transversality and projections

In the paper [4] Montaldi proves a powerful transversality result which has a number of applications to the study of the generic geometry of submanifolds of Euclidean or projective space. As a typical application one can consider the contact of surfaces with spheres in Euclidean 3-space and describe the singularities that occur generically and, consequently, the local structure of the focal and symmetry set of such surfaces. The relevant transversality result here was first established by Looijenga (see [7]). Montaldi's result not only covers this as a special case, it subsumes all known transversality theorems of this type. In this note we show how rather trivial extensions of Montalidi's Theorem can be useful in discussing the differential geometry of non-generic submanifolds of Euclidean and projective space. Such submanifolds occur naturally in families, and the sort of hypotheses one has to impose to obtain useful results are extremely mild. Roughly speaking one has to rule out only embeddings of infinite codimension, that is those which satisfy an infinite number of (degeneracy) conditions. As an application we shall give a generalisation, and incidentally a new proof, of a duality result of M. C. Romero-Fuster and the author in [2]. The original result can be viewed as a variant of a theorem of O. P. Shcherbak in [6] which deal with the case of central projections in real projective space. We also generalise some of Shcherbak's results in the final section of this paper.