Isomorphic Quartic K3 Surfaces in the View of Cremona and Projective Transformations

We show that there is a pair of smooth complex quartic K3 surfaces S-1 and S-2 in 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 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 Janos Kollar.