Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U ⊂ S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show ...

Duality between N=1 supersymmetric gauge theories(Seiberg’s duality) is geometrized, in the framework of AdS/CFT correspondences. It is shown that Seiberg’s duality corresponds to monodromy of wrapped D5 branes on the homology cycles of a generalized conifold where D3 branes are located. The celebrated Ñc = Nf −Nc, Ñf = Nf rule is reproduced and a braid group structure in a sequence of dualitie...