# Birational geometry of rational quartic surfaces

## TL;DR

This paper investigates the Cremona equivalence of rational quartic surfaces in projective space, proving that all such surfaces are Cremona equivalent to a plane, thus advancing understanding in birational geometry.

## Contribution

It establishes that all rational quartic surfaces are Cremona equivalent to a plane, providing a significant result in the classification of rational surfaces.

## Key findings

- All rational quartic surfaces are Cremona equivalent to a plane.
- The study clarifies the Cremona equivalence problem for rational surfaces.
- Advances understanding of birational transformations in algebraic geometry.

## Abstract

Two birational subvarieties of P^n are called Cremona equivalent if there is a Cremona modification of P^n mapping one to the other. If the codimension of the varieties is at least 2 then they are always Cremona Equivalent. For divisors the question is much more subtle and a general answer is unknown. In this paper I study the case of rational quartic surfaces and prove that they are all Cremona equivalent to a plane.