Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations

TL;DR

The paper constructs examples of smooth quartic K3 surfaces in projective 3-space that are isomorphic but not Cremona equivalent, and vice versa, highlighting nuanced differences in their geometric and Cremona classifications.

Contribution

It provides explicit examples demonstrating that isomorphic K3 surfaces can differ in Cremona and projective equivalences, clarifying the relationship between these classifications.

Findings

Existence of isomorphic K3 surfaces not Cremona isomorphic.

Existence of Cremona isomorphic K3 surfaces not projectively isomorphic.

Explicit geometric constructions of these examples.

Abstract

We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.