SUPERFICIAL FIBRES OF GENERIC PROJECTIONS

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.

Fibers of generic projections

Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

SURFACE LINKS AND THEIR GENERIC PLANAR PROJECTIONS

We often study surface links in 4-space by using their projections into 3-space, or their broken surface diagrams. It is well-known that a broken surface diagram recovers the given surface link. In this paper, we study surface links in 4-space by using their generic projections into the plane. These projections have fold points and cusps as their singularities in general. We consider the question whether such a generic planar projection can recover the given surface link. We introduce the notion of banded braids, and show that a generic planar projection together with banded braids associated to the segments of the fold curve image can recover the given surface link. As an application, we give a new proof to the Whitney congruence concerning the normal Euler number of surface links.

3-FOLD SYMMETRIC PRODUCTS OF CURVES AS HYPERBOLIC HYPERSURFACES IN ℙ4

We construct new examples of Kobayashi hyperbolic hypersurfaces in ℙ4. They are generic projections of the triple symmetric product V = C (3) of a generic genus g ≥ 6 curve C, smoothly embedded in ℙ7.