Department of MathematicsIn this talk I consider elliptic fourfolds with non-minimal singularities in codimension two in the base.These sectors admit an F-theory interpretation as compactifications of 6D superconformal matter on a Riemann surface. We consider resolutions of those non-minimal singularities which sacrifice flatness of the fibration and naturally leads to threeform cohomology in the fourfold which can be expressed in terms of the 6D data. In addition we consider Higgs and Tensor branch type of conifold transition that remove those codimension two non-flat fibers. The former one pushes the non-flat fiber one codimension lower and leaves non-perturbative four-point matter coupling as a remnant. The second transition removes those fibers via a base blow-up which keeps the Euler number of the fourfold
invariant.