# Ramification divisors of general projections

## TL;DR

This paper investigates how ramification divisors of projections of smooth projective varieties behave, showing they vary maximally in families, and explores the map from projections to these divisors, revealing its dominance and degree in specific cases.

## Contribution

It demonstrates the maximal variation of ramification divisors in large families and analyzes the map from projections to divisors, including its dominance and degree for certain varieties.

## Key findings

- Ramification divisors vary in maximal families for many varieties.
- The map from projections to ramification divisors is dominant for most varieties of minimal degree.
- The degree of this map relates to Catalan numbers and torsion points on elliptic curves.

## Abstract

We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, we study the map that associates to a linear projection its ramification divisor. We show that this map is dominant for most (but not all!) varieties of minimal degree, using (linked) limit linear series of higher rank. We find the degree of this map in some cases, extending the classical appearance of Catalan numbers in the geometry of rational normal curves, and give a geometric explanation of its fibers in terms of torsion points of naturally occurring elliptic curves in the case of the Veronese surface and the quartic rational surface scroll.