We consider a general fibre of given length in a generic projection of a variety. Under the assumption that the fibre is of local embedding dimension 2 or less, an assumption which can be checked in many cases, we prove that the fibre is reduced and its image on the projected variety is an ordinary multiple point.

In his celebrated paper ‘Generic Projections’, Mather obtained the key result that a generic projection to an affine subspace of a smooth submanifold in Euclidean space is jet-transverse to any ‘modular’ submanifold of (multi-) jet space. He also gave an explicit stratification by modular submanifolds, and used it to conclude that the projection, if in the nice dimensions, is generically (C∞-) stable. In this article, we extend the result to the semi-nice dimensions (where only C0-stability is obtained), using the stratification given in our book. We first recall the definitions of the nice and semi-nice dimensions, review the main known results which involve them, and proceed to the statement of our main results. Next we discuss the condition of modularity, and present a number of methods for establishing modularity of particular strata. Finally, we show that all the strata needed for the main result are covered by these methods.

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If $\cal C$ is the conductor of X over X′, and $D = Sing(X') ⊂ X'$, $Δ ⊂ X$ are the subschemes defined by $\cal C$, then D and Δ are shown to be complete intersections at these points, provided that $\cal C$ has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when $\cal C$ is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.