Fibers of Generic Projections

TL;DR

This paper introduces a new invariant for fibers of general linear projections of smooth projective varieties, providing bounds on fiber degrees and secant lines, extending classical results and improving understanding of fiber structure.

Contribution

It defines a novel fiber invariant that bounds fiber degrees and extends secant line bounds, generalizing classical results for higher dimensions.

Findings

New invariant bounds fiber degrees by n/c+1

Bound on the sum of degrees of certain fiber components

Sharp bounds on subvarieties swept out by secant lines

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".