Abstract. Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decades Cremona equivalence has been investigated widely and we have now a complete theory for non divisorial reduced schemes. The case of irreducible divisors is completely different and not much is known beside the case of plane curves and few classes of surfaces. In particular, for plane curves it is a classical result that an irreducible plane curve is Cremona equivalent to a line if and only if its log-Kodaira dimension is negative. This can be interpreted as the log version of Castelnuovo rationality criterion for surfaces. One expects that a similar result for surfaces in projective space should not be true, as it is false the generalization in higher dimension of Castelnuovo's Rationality Theorem.
In this talk I will provide a gentle introduction to the state of the art, recall the, somewhat surprising, result of reduced schemes in codimension 2 and provide an example of a surface in the projective space with negative log-Kodaira dimension which is not Cremona equivalent to a plane, this can be thought of as sort of log Iskovkikh-Manin, Clemens-Griffith, Artin-Mumfurd example.