On birational properties of smooth codimension two determinantal varieties

We show that a smooth arithmetically Cohen-Macaulay variety X, of codimension 2 in P-n if 3 <= n <= 5 and general if n > 3, admits a morphism onto a hypersurface of degree (n+1) in Pn-1 with, at worst, double points; moreover, this morphism comes from a (global) Cremona transformation which induces, by restriction to X, an isomorphism in codimension 1. We deduce that two such varieties are birationally equivalent via a Cremona transformation if and only if they are isomorphic.