• abelian with matter is the trouble

Compare with pure abelian gauge theory, the Coulomb branch of general cases needs two direction modifications, with matter representations, and with non-abelian gauge group.

In the abelian case, how do we deal with matter? We specify the multiplication table for the 'monopole operators', a vector space basis over the base coefficient ring. It is a bit brutal force.

How do we do the affine blow-up? Well, you pick two hypersurfaces in the total space, you blow-up their intersections, then remove the proper transform of one divisor.