Fibers of Generic Projections

Abstract

Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails for n >> 0. We describe a new invariant of the fiber that agrees with the degree in many cases and is always bounded by n/c+1. This implies, 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 <= n/c+1 and this formula can be strengthened a little further. Our method also gives a sharp bound on the subvariety of P^r 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 Ziv Ran's "Dimension+2 Secant Lemma".

Proof of the main theorem simplified and new examples added

References in corpus (2)

• Hilbert schemes of 8 points
• Powers of Ideals and Fibers of Morphisms

Cited by in corpus (9)

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• Equivalent birational embeddings II: divisors
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• On the smooth locus of aligned Hilbert schemes: the k-secant lemma and the general projection theorem
• Powers of Ideals and Fibers of Morphisms
• On the degree of algebraic cycles on hypersurfaces
• Graded mapping cone theorem, multisecants and syzygies
• On the size and local equations of fibres of general projections
• Fibers of Projections and Submodules of Deformations