Return to Revisions

Jim's example is good; one can easily construct examples by using elliptic fibrations. One can also make $C_1$ and $C_2$ irreducible smooth curves. Let $$S:=X\times E\rightarrow X$$ be a trivial elliptic fibration with $E$ an elliptic curve. Construct a branched double cover $Y\rightarrow X$ whose branch locus is two points. The base change of $S\rightarrow X$ will be a surface $$S'\rightarrow Y$$ with double fibers over the branch points of $Y\rightarrow X$. The difference of the set-theoretic fibers over these branch points is $2$-torsion in homology.