On Subfields of the Function Field of a General Surface in ℙ3

Abstract

In this paper, we study birational immersions from a very general smooth plane curve to a nonrational surface with pg =q =0 to treat dominant rational maps from a very general surface X of degree ≥ 5 in ℙ3 to smooth projective surfaces Y. Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from X to Y unless Y is rational or Y is birational to X.