The Minimal Cremona Degree of Quartic Surfaces

Two birational projective varieties in ℙn are Cremona Equivalent if there is a birational modification of ℙn mapping one onto the other. The minimal Cremona degree of X⊂ℙn 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 ℙ3. This allows me to show that any quartic surface of elliptic ruled type has nontrivial stabilizers in the Cremona group.