Fibers of projections and submodules of deformations

Let be a smooth projective variety of dimension n over an algebraically closed field k of characteristic zero, and let be a general linear projection. In this note we introduce some new ways of bounding the complexity of the fibers of π. Our ideas are closely related to the groundbreaking work of John Mather, and we explain a simple proof of his result [1973] bounding the Thom-Boardman invariants of π as a special case. This subject was studied classically for small n. In our situation the map π will be finite and generically one-to-one, so we are asking for bounds on the complexity of finite schemes, and the degree of the scheme is the obvious invariant. Consider, for simplicity, the case c = 1. It is well-known that the maximal degree of the fiber of a general projection of a curve to the plane is 2, and that the maximal degree of a fiber of a general projection of a smooth surface to three-space is 3. These results were extended to higher dimension and more general ground fields at the expense of strong hypotheses on the structure of the fibers by Kleiman, Roberts, Ran and others. In characteristic zero, the most striking results are those of John Mather. In the case c = 1 and he proved that a general projection it would be a stable map, and as a consequence he was able to show that, in this case, the fibers of π have degree. More generally, in case, or and, he showed that the degree of any fiber of π is bounded by. He also proved that for any n and c, the number of distinct points in any fiber is bounded by n/c + 1; this is a special case of his result bounding the Thom-Boardman invariants.