The main result of this paper is as follows: let $X,Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_2\left(X\right)=b_2\left(Y\right)=1$. If $Y$ is not a projective space, then the degree of a morphism $f:X\to Y$ is bounded in terms of discrete invariants of $X$ and $Y$. Moreover, suppose that $X$ and $Y$ are smooth projective $n$-dimensional with cyclic Néron-Severi groups. If $c_1\left(Y\right)=0$, then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional...

We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.