Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$.
Let $B\to A$ be a proper birational surjective morphism with $B$ smooth projective surface such that $B\to A$ is an isomorphism over $U$, and such that $B\setminus U$ is a divisor, say $D$. (In other words, $B\to A$ is obtained by blowing up $S$ in $A$ in an appropriate fashion.)
Why is $\omega_B + D$ not big? Is it trivial?
