  The global Seiberg-Witten (SW) geometries for rank two theories with eight
supercharges are studied. The theory is deformed generically so that there are
only simplest $I_1$ or $\tilde{I}_1$ singularities on the Coulomb branch, which
then geometrically gives the so-called Lefchetz pencils, The local singularity
was shown to be determined by the conjugacy class of mapping class group (MCG);
The global study is then reduced to the questions about MCG: a) Find the
factorization of the MCG element of the singular fiber into positive products
of Dehn twists (which gives the $I_1$ singularity or $\tilde{I}_1$
singularity); b) Find the factorization of identity element in terms of Dehn
twists. We solved above two MCG problems for most rank two theories.The results
are very helpful in determining IR physics for all vacua of 4d SCFTs. Our
approach is combinatorial and many aspects can be straightforwardly generalized
to the study of higher rank theory.
