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 X⇢Pn. It is a well-known fact that given a product X×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⊂P3 of degree d≥5, and we prove that irr(S×Pm)=irr(S) for any integer m≥0, whereas irr(Y)