On the size and local equations of fibres of general projections

TL;DR

This paper investigates the fibers of general projections of smooth projective varieties, showing they are bounded in size and locally defined by simple equations, with implications for algebraic geometry.

Contribution

It establishes bounds on fiber sizes and describes their local equations for general projections of smooth projective varieties, extending understanding of their geometric structure.

Findings

Fibers have total length asymptotically bounded by 2^{√n}+1.

Fibers are locally defined by linear and quadratic equations.

Results apply to general birational projections from P^{n+c} to P^m.

Abstract

For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1} $ and the fibres are locally defined by linear and quadratic equations.