  In this paper we study birational immersions from a very general smooth plane
curve to a non-rational surface with $p_g=q=0$ to treat dominant rational maps
from a very general surface $X$ of degree$\geq 5$ in ${\mathbb P}^3$ 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 $X$.
