The minimal Cremona degree of quartic surfaces

TL;DR

This paper computes the minimal Cremona degree of quartic surfaces in projective 3-space, revealing new insights into their birational properties and symmetries within the Cremona group.

Contribution

It provides the first explicit calculation of the minimal Cremona degree for quartic surfaces, a previously subtle problem in algebraic geometry.

Findings

Quartic surfaces of elliptic ruled type have non-trivial stabilizers in the Cremona group.

The minimal Cremona degree for these surfaces is explicitly determined.

The results deepen understanding of Cremona equivalence for divisors.

Abstract

Two birational projective varieties in $P^n$ are Cremona Equivalent if there is a birational modification of $P^n$ mapping one onto the other. The minimal Cremona degree of $X\subset P^n$ is the minimal integer among all degrees of varieties that are Cremona Equivalent to $X$. The Cremona Equivalence and the minimal Cremona degree is well understood for subvarieties of codimension at least $2$ while both are in general very subtle questions for divisors. In this note I compute the minimal Cremona degree of quartic surfaces in $P^3$. This allows me to show that any quartic surface of elliptic ruled type has non trivial stabilizers in the Cremona group.