On irrationality of surfaces in P3

The degree of irrationality irr(X) of a n-dimensional complex projective variety X is the least degree of a dominant rational map from X to the n-dimensional projective space Pn. It is a well-known fact that given a product Xx(Pm) or a n-dimensional variety Y dominating X, their degrees of irrationality may be smaller than the degree of irrationality of X. In this paper, we focus on smooth surfaces S in the 3-dimensional projective space ‚P3 of degree d>4, and we prove that irr(Sx(Pm))=irr(S) for any integer m>0, whereas irr(Y)