Let us prove that any non-simply connected surface without curves of negative self-intersection provides a counterexample
This answer is completely rewritten. This includes Abelian surfaces and generic products of curves of higher genusis not an actual answer but a thought related to the question.
ProofI decided to leave it hear since it is short. Suppose
Note first that $X$ if there is a regular map from a surface with isolated singularities in $X$ to $\mathbb CP^3$. Then the sinuglarities of the surfaces are normal. Hence the minimal resolution of these P^3$whose image has only isolated singularitiesmust have , then $X$ has curves with negative self-intersections (that project to singularity)self-intersection. So In particular, if the resolution does not have $X$ has no such curves, then its image in $X$ \mathbb P^3$ is non-singularsmooth.It follows that
Now, suppose we have a surface $X$ has trivial fundamental group.
Correction. I see now that with isolated singularities in order for this reasoning to work one has to guarantee that on $\mathbb CP^3$,say of general type and consider the question:
Question. Let $X'$ be the minimal resolution of singularities of on $X$ there are no rational X$.Can we say something about $-1$-curves. I wonder X$ if this holds indeed provided the degree of $X$ is high enough. Or maybe one can "classify" surfaces in X'$ contains rational $\mathbb CP^3$ for which this statement fails. So I still believe this idea can work out...-1$ curves?