Equivalent birational embeddings

TL;DR

This paper proves that any two birational embeddings of a projective variety into projective space of dimension at least r+2 are equivalent through Cremona transformations, establishing a fundamental relation between such embeddings.

Contribution

It establishes that all birational embeddings of a projective variety into sufficiently high-dimensional projective space are Cremona equivalent.

Findings

Any two birational embeddings in ^n, n e2 r+2, are Cremona equivalent.

The result applies to varieties over algebraically closed fields.

Provides a unifying view of birational embeddings via Cremona transformations.

Abstract

Let $X$ be a projective variety of dimension $r$ over an algebraically closed field. It is proven that two birational embeddings of $X$ in $\P^n$, with $n\geq r+2$ are equivalent up to Cremona transformations of $\P^n$.