### Abstract

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.

Original language | English |
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Pages (from-to) | 183-194 |

Journal | Journal of the London Mathematical Society |

Volume | 49 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1994 |

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## Cite this

Bruce, J. W. (1994). Generic geometry, transversality and projections.

*Journal of the London Mathematical Society*,*49*(1), 183-194. https://doi.org/10.1112/jlms/49.1.183