Equivalent birational embeddings.

Let X be a projective variety of dimension r. We want to understand when two birational embeddings of the same variety are equivalent up to a Cremona transformation of the projective space, in this case we say that they are Cremona equivalent. It is proven that two birational embeddings of X in Pn with n >= r + 2 are Cremona equivalent. To do this, it is produced a chain of Cremona transformations that modify the linear systems giving the two embeddings one into the other. This is done by looking at the two birational embeddings as different projections of a common embedding. On the other hand, if n = r + 1, there are birationally divisors that are not Cremona equivalent. The case of plane curves is studied in details. Let C be an irreducible and reduced plane curve of arbitrary genus. It is proven that the curve C is birational to either a line; either a curve C, where the log pair (P2,3/dC)has canonical singularities, the log canonical divisor nef and Kodaira dimension k = 0; or a curve C ~ aC0 + bf Fa, where the log pair (Fa,2/aC) has canonical singularities and terminal singularities in a neighborhood of the exceptional curve C0 Fa, the log canonical divisor nef and Kodaira dimension k <= 1. Finally, it is used the theory of &–minimal models to under- stand whether a rational, irreducible and reduced curve is Cremona equivalent to a line.